/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 944 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QReductionProof [EQUIVALENT, 25 ms]
        (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) UsableRulesProof [EQUIVALENT, 0 ms]
        (14) QDP
        (15) QReductionProof [EQUIVALENT, 24 ms]
        (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) QDP
        (20) UsableRulesProof [EQUIVALENT, 0 ms]
        (21) QDP
        (22) QReductionProof [EQUIVALENT, 29 ms]
        (23) QDP
        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (25) YES
    (26) QDP
        (27) UsableRulesProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) QReductionProof [EQUIVALENT, 26 ms]
        (30) QDP
        (31) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (32) YES
    (33) QDP
        (34) UsableRulesProof [EQUIVALENT, 0 ms]
        (35) QDP
        (36) QReductionProof [EQUIVALENT, 31 ms]
        (37) QDP
        (38) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (39) YES
    (40) QDP
        (41) UsableRulesProof [EQUIVALENT, 0 ms]
        (42) QDP
        (43) QReductionProof [EQUIVALENT, 28 ms]
        (44) QDP
        (45) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (46) YES
    (47) QDP
        (48) UsableRulesProof [EQUIVALENT, 0 ms]
        (49) QDP
        (50) QReductionProof [EQUIVALENT, 15 ms]
        (51) QDP
        (52) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (53) YES
    (54) QDP
        (55) UsableRulesProof [EQUIVALENT, 0 ms]
        (56) QDP
        (57) QReductionProof [EQUIVALENT, 6 ms]
        (58) QDP
        (59) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (60) YES
    (61) QDP
        (62) UsableRulesProof [EQUIVALENT, 0 ms]
        (63) QDP
        (64) QReductionProof [EQUIVALENT, 8 ms]
        (65) QDP
        (66) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (67) YES
    (68) QDP
        (69) UsableRulesProof [EQUIVALENT, 0 ms]
        (70) QDP
        (71) QReductionProof [EQUIVALENT, 10 ms]
        (72) QDP
        (73) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (74) YES
    (75) QDP
        (76) UsableRulesProof [EQUIVALENT, 0 ms]
        (77) QDP
        (78) QReductionProof [EQUIVALENT, 11 ms]
        (79) QDP
        (80) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (81) YES
    (82) QDP
        (83) UsableRulesProof [EQUIVALENT, 0 ms]
        (84) QDP
        (85) QReductionProof [EQUIVALENT, 6 ms]
        (86) QDP
        (87) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (88) YES
    (89) QDP
        (90) UsableRulesProof [EQUIVALENT, 0 ms]
        (91) QDP
        (92) QReductionProof [EQUIVALENT, 6 ms]
        (93) QDP
        (94) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (95) YES
    (96) QDP
        (97) UsableRulesProof [EQUIVALENT, 0 ms]
        (98) QDP
        (99) QReductionProof [EQUIVALENT, 7 ms]
        (100) QDP
        (101) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (102) YES
    (103) QDP
        (104) UsableRulesProof [EQUIVALENT, 0 ms]
        (105) QDP
        (106) QReductionProof [EQUIVALENT, 0 ms]
        (107) QDP
        (108) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (109) YES
    (110) QDP
        (111) UsableRulesProof [EQUIVALENT, 0 ms]
        (112) QDP
        (113) QReductionProof [EQUIVALENT, 0 ms]
        (114) QDP
        (115) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (116) YES
    (117) QDP
        (118) UsableRulesProof [EQUIVALENT, 0 ms]
        (119) QDP
        (120) QReductionProof [EQUIVALENT, 0 ms]
        (121) QDP
        (122) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (123) YES
    (124) QDP
        (125) UsableRulesProof [EQUIVALENT, 0 ms]
        (126) QDP
        (127) QReductionProof [EQUIVALENT, 0 ms]
        (128) QDP
        (129) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (130) YES
    (131) QDP
        (132) UsableRulesProof [EQUIVALENT, 0 ms]
        (133) QDP
        (134) QReductionProof [EQUIVALENT, 0 ms]
        (135) QDP
        (136) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (137) YES
    (138) QDP
        (139) UsableRulesProof [EQUIVALENT, 0 ms]
        (140) QDP
        (141) QReductionProof [EQUIVALENT, 0 ms]
        (142) QDP
        (143) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (144) YES
    (145) QDP
        (146) UsableRulesProof [EQUIVALENT, 0 ms]
        (147) QDP
        (148) QReductionProof [EQUIVALENT, 0 ms]
        (149) QDP
        (150) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (151) YES
    (152) QDP
        (153) UsableRulesProof [EQUIVALENT, 0 ms]
        (154) QDP
        (155) QReductionProof [EQUIVALENT, 0 ms]
        (156) QDP
        (157) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (158) YES
    (159) QDP
        (160) UsableRulesProof [EQUIVALENT, 0 ms]
        (161) QDP
        (162) QReductionProof [EQUIVALENT, 0 ms]
        (163) QDP
        (164) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (165) YES
    (166) QDP
        (167) UsableRulesProof [EQUIVALENT, 0 ms]
        (168) QDP
        (169) QReductionProof [EQUIVALENT, 0 ms]
        (170) QDP
        (171) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (172) YES
    (173) QDP
        (174) UsableRulesProof [EQUIVALENT, 0 ms]
        (175) QDP
        (176) QReductionProof [EQUIVALENT, 0 ms]
        (177) QDP
        (178) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (179) YES
    (180) QDP
        (181) UsableRulesProof [EQUIVALENT, 0 ms]
        (182) QDP
        (183) QReductionProof [EQUIVALENT, 11 ms]
        (184) QDP
        (185) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (186) YES
    (187) QDP
        (188) UsableRulesProof [EQUIVALENT, 0 ms]
        (189) QDP
        (190) QReductionProof [EQUIVALENT, 0 ms]
        (191) QDP
        (192) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (193) YES
    (194) QDP
        (195) UsableRulesProof [EQUIVALENT, 0 ms]
        (196) QDP
        (197) QReductionProof [EQUIVALENT, 0 ms]
        (198) QDP
        (199) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (200) YES
    (201) QDP
        (202) UsableRulesProof [EQUIVALENT, 0 ms]
        (203) QDP
        (204) QReductionProof [EQUIVALENT, 10 ms]
        (205) QDP
        (206) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (207) YES
    (208) QDP
        (209) UsableRulesProof [EQUIVALENT, 0 ms]
        (210) QDP
        (211) QReductionProof [EQUIVALENT, 0 ms]
        (212) QDP
        (213) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (214) YES
    (215) QDP
        (216) UsableRulesProof [EQUIVALENT, 0 ms]
        (217) QDP
        (218) QReductionProof [EQUIVALENT, 0 ms]
        (219) QDP
        (220) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (221) YES
    (222) QDP
        (223) UsableRulesProof [EQUIVALENT, 0 ms]
        (224) QDP
        (225) QReductionProof [EQUIVALENT, 0 ms]
        (226) QDP
        (227) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (228) YES
    (229) QDP
        (230) UsableRulesProof [EQUIVALENT, 0 ms]
        (231) QDP
        (232) QReductionProof [EQUIVALENT, 0 ms]
        (233) QDP
        (234) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (235) YES
    (236) QDP
        (237) UsableRulesProof [EQUIVALENT, 0 ms]
        (238) QDP
        (239) QReductionProof [EQUIVALENT, 0 ms]
        (240) QDP
        (241) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (242) YES
    (243) QDP
        (244) UsableRulesProof [EQUIVALENT, 0 ms]
        (245) QDP
        (246) QReductionProof [EQUIVALENT, 0 ms]
        (247) QDP
        (248) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (249) YES
    (250) QDP
        (251) UsableRulesProof [EQUIVALENT, 0 ms]
        (252) QDP
        (253) QReductionProof [EQUIVALENT, 0 ms]
        (254) QDP
        (255) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (256) YES
    (257) QDP
        (258) UsableRulesProof [EQUIVALENT, 0 ms]
        (259) QDP
        (260) QReductionProof [EQUIVALENT, 0 ms]
        (261) QDP
        (262) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (263) YES
    (264) QDP
        (265) UsableRulesProof [EQUIVALENT, 0 ms]
        (266) QDP
        (267) QReductionProof [EQUIVALENT, 0 ms]
        (268) QDP
        (269) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (270) YES
    (271) QDP
        (272) UsableRulesProof [EQUIVALENT, 0 ms]
        (273) QDP
        (274) QReductionProof [EQUIVALENT, 0 ms]
        (275) QDP
        (276) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (277) YES
    (278) QDP
        (279) UsableRulesProof [EQUIVALENT, 0 ms]
        (280) QDP
        (281) QReductionProof [EQUIVALENT, 0 ms]
        (282) QDP
        (283) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (284) YES
    (285) QDP
        (286) UsableRulesProof [EQUIVALENT, 0 ms]
        (287) QDP
        (288) QReductionProof [EQUIVALENT, 0 ms]
        (289) QDP
        (290) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (291) YES
    (292) QDP
        (293) UsableRulesProof [EQUIVALENT, 0 ms]
        (294) QDP
        (295) QReductionProof [EQUIVALENT, 0 ms]
        (296) QDP
        (297) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (298) YES
    (299) QDP
        (300) UsableRulesProof [EQUIVALENT, 0 ms]
        (301) QDP
        (302) QReductionProof [EQUIVALENT, 0 ms]
        (303) QDP
        (304) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (305) YES
    (306) QDP
        (307) UsableRulesProof [EQUIVALENT, 0 ms]
        (308) QDP
        (309) QReductionProof [EQUIVALENT, 0 ms]
        (310) QDP
        (311) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (312) YES
    (313) QDP
        (314) UsableRulesProof [EQUIVALENT, 0 ms]
        (315) QDP
        (316) QReductionProof [EQUIVALENT, 0 ms]
        (317) QDP
        (318) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (319) YES
    (320) QDP
        (321) UsableRulesProof [EQUIVALENT, 0 ms]
        (322) QDP
        (323) QReductionProof [EQUIVALENT, 0 ms]
        (324) QDP
        (325) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (326) YES
    (327) QDP
        (328) UsableRulesProof [EQUIVALENT, 0 ms]
        (329) QDP
        (330) QReductionProof [EQUIVALENT, 0 ms]
        (331) QDP
        (332) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (333) YES
    (334) QDP
        (335) UsableRulesProof [EQUIVALENT, 0 ms]
        (336) QDP
        (337) QReductionProof [EQUIVALENT, 0 ms]
        (338) QDP
        (339) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (340) YES
    (341) QDP
        (342) UsableRulesProof [EQUIVALENT, 0 ms]
        (343) QDP
        (344) QReductionProof [EQUIVALENT, 0 ms]
        (345) QDP
        (346) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (347) YES
    (348) QDP
        (349) UsableRulesProof [EQUIVALENT, 0 ms]
        (350) QDP
        (351) QReductionProof [EQUIVALENT, 0 ms]
        (352) QDP
        (353) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (354) YES
    (355) QDP
        (356) UsableRulesProof [EQUIVALENT, 0 ms]
        (357) QDP
        (358) QReductionProof [EQUIVALENT, 0 ms]
        (359) QDP
        (360) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (361) YES
    (362) QDP
        (363) UsableRulesProof [EQUIVALENT, 0 ms]
        (364) QDP
        (365) QReductionProof [EQUIVALENT, 0 ms]
        (366) QDP
        (367) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (368) YES
    (369) QDP
        (370) UsableRulesProof [EQUIVALENT, 0 ms]
        (371) QDP
        (372) QReductionProof [EQUIVALENT, 0 ms]
        (373) QDP
        (374) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (375) YES
    (376) QDP
        (377) UsableRulesProof [EQUIVALENT, 0 ms]
        (378) QDP
        (379) QReductionProof [EQUIVALENT, 0 ms]
        (380) QDP
        (381) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (382) YES
    (383) QDP
        (384) UsableRulesProof [EQUIVALENT, 0 ms]
        (385) QDP
        (386) QDPOrderProof [EQUIVALENT, 2510 ms]
        (387) QDP
        (388) QDPOrderProof [EQUIVALENT, 2695 ms]
        (389) QDP
        (390) QDPOrderProof [EQUIVALENT, 2432 ms]
        (391) QDP
        (392) QDPOrderProof [EQUIVALENT, 338 ms]
        (393) QDP
        (394) QDPOrderProof [EQUIVALENT, 359 ms]
        (395) QDP
        (396) QDPOrderProof [EQUIVALENT, 2283 ms]
        (397) QDP
        (398) QDPOrderProof [EQUIVALENT, 311 ms]
        (399) QDP
        (400) QDPOrderProof [EQUIVALENT, 295 ms]
        (401) QDP
        (402) QDPOrderProof [EQUIVALENT, 2738 ms]
        (403) QDP
        (404) DependencyGraphProof [EQUIVALENT, 0 ms]
        (405) QDP
        (406) QDPOrderProof [EQUIVALENT, 2336 ms]
        (407) QDP
        (408) DependencyGraphProof [EQUIVALENT, 0 ms]
        (409) QDP
        (410) QDPOrderProof [EQUIVALENT, 277 ms]
        (411) QDP
        (412) QDPOrderProof [EQUIVALENT, 2294 ms]
        (413) QDP
        (414) QDPOrderProof [EQUIVALENT, 301 ms]
        (415) QDP
        (416) QDPOrderProof [EQUIVALENT, 2192 ms]
        (417) QDP
        (418) QDPOrderProof [EQUIVALENT, 217 ms]
        (419) QDP
        (420) QDPOrderProof [EQUIVALENT, 246 ms]
        (421) QDP
        (422) QDPOrderProof [EQUIVALENT, 2327 ms]
        (423) QDP
        (424) DependencyGraphProof [EQUIVALENT, 0 ms]
        (425) QDP
        (426) QDPOrderProof [EQUIVALENT, 2419 ms]
        (427) QDP
        (428) DependencyGraphProof [EQUIVALENT, 0 ms]
        (429) QDP
        (430) QDPOrderProof [EQUIVALENT, 2289 ms]
        (431) QDP
        (432) QDPOrderProof [EQUIVALENT, 1906 ms]
        (433) QDP
        (434) QDPOrderProof [EQUIVALENT, 2337 ms]
        (435) QDP
        (436) DependencyGraphProof [EQUIVALENT, 0 ms]
        (437) QDP
        (438) QDPOrderProof [EQUIVALENT, 1896 ms]
        (439) QDP
        (440) QDPOrderProof [EQUIVALENT, 2230 ms]
        (441) QDP
        (442) QDPOrderProof [EQUIVALENT, 1810 ms]
        (443) QDP
        (444) QDPOrderProof [EQUIVALENT, 2109 ms]
        (445) QDP
        (446) QDPOrderProof [EQUIVALENT, 2002 ms]
        (447) QDP
        (448) QDPOrderProof [EQUIVALENT, 184 ms]
        (449) QDP
        (450) QDPOrderProof [EQUIVALENT, 2101 ms]
        (451) QDP
        (452) QDPOrderProof [EQUIVALENT, 2144 ms]
        (453) QDP
        (454) DependencyGraphProof [EQUIVALENT, 0 ms]
        (455) QDP
        (456) QDPOrderProof [EQUIVALENT, 2104 ms]
        (457) QDP
        (458) DependencyGraphProof [EQUIVALENT, 0 ms]
        (459) QDP
        (460) QDPOrderProof [EQUIVALENT, 194 ms]
        (461) QDP
        (462) QDPOrderProof [EQUIVALENT, 2024 ms]
        (463) QDP
        (464) QDPOrderProof [EQUIVALENT, 2156 ms]
        (465) QDP
        (466) DependencyGraphProof [EQUIVALENT, 0 ms]
        (467) QDP
        (468) QDPOrderProof [EQUIVALENT, 2054 ms]
        (469) QDP
        (470) DependencyGraphProof [EQUIVALENT, 0 ms]
        (471) QDP
        (472) QDPOrderProof [EQUIVALENT, 176 ms]
        (473) QDP
        (474) QDPOrderProof [EQUIVALENT, 3358 ms]
        (475) QDP
        (476) QDPOrderProof [EQUIVALENT, 185 ms]
        (477) QDP
        (478) QDPOrderProof [EQUIVALENT, 2754 ms]
        (479) QDP
        (480) QDPOrderProof [EQUIVALENT, 1758 ms]
        (481) QDP
        (482) QDPOrderProof [EQUIVALENT, 1923 ms]
        (483) QDP
        (484) QDPOrderProof [EQUIVALENT, 195 ms]
        (485) QDP
        (486) QDPOrderProof [EQUIVALENT, 2239 ms]
        (487) QDP
        (488) QDPOrderProof [EQUIVALENT, 188 ms]
        (489) QDP
        (490) QDPOrderProof [EQUIVALENT, 188 ms]
        (491) QDP
        (492) DependencyGraphProof [EQUIVALENT, 0 ms]
        (493) QDP
        (494) QDPOrderProof [EQUIVALENT, 2170 ms]
        (495) QDP
        (496) QDPOrderProof [EQUIVALENT, 1896 ms]
        (497) QDP
        (498) QDPOrderProof [EQUIVALENT, 2440 ms]
        (499) QDP
        (500) QDPOrderProof [EQUIVALENT, 1523 ms]
        (501) QDP
        (502) QDPOrderProof [EQUIVALENT, 1694 ms]
        (503) QDP
        (504) QDPOrderProof [EQUIVALENT, 1491 ms]
        (505) QDP
        (506) QDPOrderProof [EQUIVALENT, 1436 ms]
        (507) QDP
        (508) QDPOrderProof [EQUIVALENT, 1455 ms]
        (509) QDP
        (510) QDPOrderProof [EQUIVALENT, 1694 ms]
        (511) QDP
        (512) QDPOrderProof [EQUIVALENT, 1563 ms]
        (513) QDP
        (514) QDPOrderProof [EQUIVALENT, 1643 ms]
        (515) QDP
        (516) QDPOrderProof [EQUIVALENT, 1525 ms]
        (517) QDP
        (518) QDPOrderProof [EQUIVALENT, 1712 ms]
        (519) QDP
        (520) QDPOrderProof [EQUIVALENT, 1286 ms]
        (521) QDP
        (522) QDPOrderProof [EQUIVALENT, 1416 ms]
        (523) QDP
        (524) QDPOrderProof [EQUIVALENT, 1306 ms]
        (525) QDP
        (526) QDPOrderProof [EQUIVALENT, 1310 ms]
        (527) QDP
        (528) QDPOrderProof [EQUIVALENT, 1445 ms]
        (529) QDP
        (530) DependencyGraphProof [EQUIVALENT, 0 ms]
        (531) QDP
        (532) QDPOrderProof [EQUIVALENT, 1366 ms]
        (533) QDP
        (534) QDPOrderProof [EQUIVALENT, 2559 ms]
        (535) QDP
        (536) QDPOrderProof [EQUIVALENT, 1471 ms]
        (537) QDP
        (538) DependencyGraphProof [EQUIVALENT, 0 ms]
        (539) QDP
        (540) QDPOrderProof [EQUIVALENT, 1396 ms]
        (541) QDP
        (542) QDPOrderProof [EQUIVALENT, 1222 ms]
        (543) QDP
        (544) QDPOrderProof [EQUIVALENT, 1231 ms]
        (545) QDP
        (546) QDPOrderProof [EQUIVALENT, 1295 ms]
        (547) QDP
        (548) QDPOrderProof [EQUIVALENT, 1260 ms]
        (549) QDP
        (550) QDPOrderProof [EQUIVALENT, 104 ms]
        (551) QDP
        (552) DependencyGraphProof [EQUIVALENT, 0 ms]
        (553) AND
            (554) QDP
                (555) TransformationProof [EQUIVALENT, 0 ms]
                (556) QDP
                (557) QDPOrderProof [EQUIVALENT, 553 ms]
                (558) QDP
                (559) PisEmptyProof [EQUIVALENT, 0 ms]
                (560) YES
            (561) QDP
                (562) UsableRulesProof [EQUIVALENT, 0 ms]
                (563) QDP
                (564) QReductionProof [EQUIVALENT, 0 ms]
                (565) QDP
                (566) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (567) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(zeros) -> MARK(cons(0, zeros))
   ACTIVE(zeros) -> CONS(0, zeros)
   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   ACTIVE(U101(tt, V1, V2)) -> U102^1(isNatKind(V1), V1, V2)
   ACTIVE(U101(tt, V1, V2)) -> ISNATKIND(V1)
   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   ACTIVE(U102(tt, V1, V2)) -> U103^1(isNatIListKind(V2), V1, V2)
   ACTIVE(U102(tt, V1, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   ACTIVE(U103(tt, V1, V2)) -> U104^1(isNatIListKind(V2), V1, V2)
   ACTIVE(U103(tt, V1, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   ACTIVE(U104(tt, V1, V2)) -> U105^1(isNat(V1), V2)
   ACTIVE(U104(tt, V1, V2)) -> ISNAT(V1)
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U105(tt, V2)) -> U106^1(isNatIList(V2))
   ACTIVE(U105(tt, V2)) -> ISNATILIST(V2)
   ACTIVE(U106(tt)) -> MARK(tt)
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U11(tt, V1)) -> U12^1(isNatIListKind(V1), V1)
   ACTIVE(U11(tt, V1)) -> ISNATILISTKIND(V1)
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   ACTIVE(U111(tt, L, N)) -> U112^1(isNatIListKind(L), L, N)
   ACTIVE(U111(tt, L, N)) -> ISNATILISTKIND(L)
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   ACTIVE(U112(tt, L, N)) -> U113^1(isNat(N), L, N)
   ACTIVE(U112(tt, L, N)) -> ISNAT(N)
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U113(tt, L, N)) -> U114^1(isNatKind(N), L)
   ACTIVE(U113(tt, L, N)) -> ISNATKIND(N)
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U114(tt, L)) -> S(length(L))
   ACTIVE(U114(tt, L)) -> LENGTH(L)
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U12(tt, V1)) -> U13^1(isNatList(V1))
   ACTIVE(U12(tt, V1)) -> ISNATLIST(V1)
   ACTIVE(U121(tt, IL)) -> MARK(U122(isNatIListKind(IL)))
   ACTIVE(U121(tt, IL)) -> U122^1(isNatIListKind(IL))
   ACTIVE(U121(tt, IL)) -> ISNATILISTKIND(IL)
   ACTIVE(U122(tt)) -> MARK(nil)
   ACTIVE(U13(tt)) -> MARK(tt)
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   ACTIVE(U131(tt, IL, M, N)) -> U132^1(isNatIListKind(IL), IL, M, N)
   ACTIVE(U131(tt, IL, M, N)) -> ISNATILISTKIND(IL)
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   ACTIVE(U132(tt, IL, M, N)) -> U133^1(isNat(M), IL, M, N)
   ACTIVE(U132(tt, IL, M, N)) -> ISNAT(M)
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   ACTIVE(U133(tt, IL, M, N)) -> U134^1(isNatKind(M), IL, M, N)
   ACTIVE(U133(tt, IL, M, N)) -> ISNATKIND(M)
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   ACTIVE(U134(tt, IL, M, N)) -> U135^1(isNat(N), IL, M, N)
   ACTIVE(U134(tt, IL, M, N)) -> ISNAT(N)
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   ACTIVE(U135(tt, IL, M, N)) -> U136^1(isNatKind(N), IL, M, N)
   ACTIVE(U135(tt, IL, M, N)) -> ISNATKIND(N)
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(U136(tt, IL, M, N)) -> CONS(N, take(M, IL))
   ACTIVE(U136(tt, IL, M, N)) -> TAKE(M, IL)
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U21(tt, V1)) -> U22^1(isNatKind(V1), V1)
   ACTIVE(U21(tt, V1)) -> ISNATKIND(V1)
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U22(tt, V1)) -> U23^1(isNat(V1))
   ACTIVE(U22(tt, V1)) -> ISNAT(V1)
   ACTIVE(U23(tt)) -> MARK(tt)
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(U31(tt, V)) -> U32^1(isNatIListKind(V), V)
   ACTIVE(U31(tt, V)) -> ISNATILISTKIND(V)
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U32(tt, V)) -> U33^1(isNatList(V))
   ACTIVE(U32(tt, V)) -> ISNATLIST(V)
   ACTIVE(U33(tt)) -> MARK(tt)
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U41(tt, V1, V2)) -> U42^1(isNatKind(V1), V1, V2)
   ACTIVE(U41(tt, V1, V2)) -> ISNATKIND(V1)
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U42(tt, V1, V2)) -> U43^1(isNatIListKind(V2), V1, V2)
   ACTIVE(U42(tt, V1, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> U44^1(isNatIListKind(V2), V1, V2)
   ACTIVE(U43(tt, V1, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   ACTIVE(U44(tt, V1, V2)) -> U45^1(isNat(V1), V2)
   ACTIVE(U44(tt, V1, V2)) -> ISNAT(V1)
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U45(tt, V2)) -> U46^1(isNatIList(V2))
   ACTIVE(U45(tt, V2)) -> ISNATILIST(V2)
   ACTIVE(U46(tt)) -> MARK(tt)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U51(tt, V2)) -> U52^1(isNatIListKind(V2))
   ACTIVE(U51(tt, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U52(tt)) -> MARK(tt)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> U62^1(isNatIListKind(V2))
   ACTIVE(U61(tt, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U62(tt)) -> MARK(tt)
   ACTIVE(U71(tt)) -> MARK(tt)
   ACTIVE(U81(tt)) -> MARK(tt)
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   ACTIVE(U91(tt, V1, V2)) -> U92^1(isNatKind(V1), V1, V2)
   ACTIVE(U91(tt, V1, V2)) -> ISNATKIND(V1)
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   ACTIVE(U92(tt, V1, V2)) -> U93^1(isNatIListKind(V2), V1, V2)
   ACTIVE(U92(tt, V1, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   ACTIVE(U93(tt, V1, V2)) -> U94^1(isNatIListKind(V2), V1, V2)
   ACTIVE(U93(tt, V1, V2)) -> ISNATILISTKIND(V2)
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   ACTIVE(U94(tt, V1, V2)) -> U95^1(isNat(V1), V2)
   ACTIVE(U94(tt, V1, V2)) -> ISNAT(V1)
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   ACTIVE(U95(tt, V2)) -> U96^1(isNatList(V2))
   ACTIVE(U95(tt, V2)) -> ISNATLIST(V2)
   ACTIVE(U96(tt)) -> MARK(tt)
   ACTIVE(isNat(0)) -> MARK(tt)
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   ACTIVE(isNat(length(V1))) -> U11^1(isNatIListKind(V1), V1)
   ACTIVE(isNat(length(V1))) -> ISNATILISTKIND(V1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNat(s(V1))) -> U21^1(isNatKind(V1), V1)
   ACTIVE(isNat(s(V1))) -> ISNATKIND(V1)
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   ACTIVE(isNatIList(V)) -> U31^1(isNatIListKind(V), V)
   ACTIVE(isNatIList(V)) -> ISNATILISTKIND(V)
   ACTIVE(isNatIList(zeros)) -> MARK(tt)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> U41^1(isNatKind(V1), V1, V2)
   ACTIVE(isNatIList(cons(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(isNatIListKind(nil)) -> MARK(tt)
   ACTIVE(isNatIListKind(zeros)) -> MARK(tt)
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> U51^1(isNatKind(V1), V2)
   ACTIVE(isNatIListKind(cons(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatIListKind(take(V1, V2))) -> U61^1(isNatKind(V1), V2)
   ACTIVE(isNatIListKind(take(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(isNatKind(0)) -> MARK(tt)
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   ACTIVE(isNatKind(length(V1))) -> U71^1(isNatIListKind(V1))
   ACTIVE(isNatKind(length(V1))) -> ISNATILISTKIND(V1)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> U81^1(isNatKind(V1))
   ACTIVE(isNatKind(s(V1))) -> ISNATKIND(V1)
   ACTIVE(isNatList(nil)) -> MARK(tt)
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(isNatList(cons(V1, V2))) -> U91^1(isNatKind(V1), V1, V2)
   ACTIVE(isNatList(cons(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatList(take(V1, V2))) -> U101^1(isNatKind(V1), V1, V2)
   ACTIVE(isNatList(take(V1, V2))) -> ISNATKIND(V1)
   ACTIVE(length(nil)) -> MARK(0)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(length(cons(N, L))) -> U111^1(isNatList(L), L, N)
   ACTIVE(length(cons(N, L))) -> ISNATLIST(L)
   ACTIVE(take(0, IL)) -> MARK(U121(isNatIList(IL), IL))
   ACTIVE(take(0, IL)) -> U121^1(isNatIList(IL), IL)
   ACTIVE(take(0, IL)) -> ISNATILIST(IL)
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   ACTIVE(take(s(M), cons(N, IL))) -> U131^1(isNatIList(IL), IL, M, N)
   ACTIVE(take(s(M), cons(N, IL))) -> ISNATILIST(IL)
   MARK(zeros) -> ACTIVE(zeros)
   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   MARK(cons(X1, X2)) -> CONS(mark(X1), X2)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(0) -> ACTIVE(0)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   MARK(U101(X1, X2, X3)) -> U101^1(mark(X1), X2, X3)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(tt) -> ACTIVE(tt)
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   MARK(U102(X1, X2, X3)) -> U102^1(mark(X1), X2, X3)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   MARK(U103(X1, X2, X3)) -> U103^1(mark(X1), X2, X3)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   MARK(U104(X1, X2, X3)) -> U104^1(mark(X1), X2, X3)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   MARK(U105(X1, X2)) -> U105^1(mark(X1), X2)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   MARK(U106(X)) -> ACTIVE(U106(mark(X)))
   MARK(U106(X)) -> U106^1(mark(X))
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> U11^1(mark(X1), X2)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   MARK(U12(X1, X2)) -> U12^1(mark(X1), X2)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   MARK(U111(X1, X2, X3)) -> U111^1(mark(X1), X2, X3)
   MARK(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   MARK(U112(X1, X2, X3)) -> U112^1(mark(X1), X2, X3)
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   MARK(U113(X1, X2, X3)) -> U113^1(mark(X1), X2, X3)
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   MARK(U114(X1, X2)) -> U114^1(mark(X1), X2)
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   MARK(s(X)) -> S(mark(X))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> LENGTH(mark(X))
   MARK(length(X)) -> MARK(X)
   MARK(U13(X)) -> ACTIVE(U13(mark(X)))
   MARK(U13(X)) -> U13^1(mark(X))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
   MARK(U121(X1, X2)) -> U121^1(mark(X1), X2)
   MARK(U121(X1, X2)) -> MARK(X1)
   MARK(U122(X)) -> ACTIVE(U122(mark(X)))
   MARK(U122(X)) -> U122^1(mark(X))
   MARK(U122(X)) -> MARK(X)
   MARK(nil) -> ACTIVE(nil)
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(U131(X1, X2, X3, X4)) -> U131^1(mark(X1), X2, X3, X4)
   MARK(U131(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   MARK(U132(X1, X2, X3, X4)) -> U132^1(mark(X1), X2, X3, X4)
   MARK(U132(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   MARK(U133(X1, X2, X3, X4)) -> U133^1(mark(X1), X2, X3, X4)
   MARK(U133(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   MARK(U134(X1, X2, X3, X4)) -> U134^1(mark(X1), X2, X3, X4)
   MARK(U134(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   MARK(U135(X1, X2, X3, X4)) -> U135^1(mark(X1), X2, X3, X4)
   MARK(U135(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   MARK(U136(X1, X2, X3, X4)) -> U136^1(mark(X1), X2, X3, X4)
   MARK(U136(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> TAKE(mark(X1), mark(X2))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> U21^1(mark(X1), X2)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U22(X1, X2)) -> U22^1(mark(X1), X2)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   MARK(U23(X)) -> U23^1(mark(X))
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> U31^1(mark(X1), X2)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   MARK(U32(X1, X2)) -> U32^1(mark(X1), X2)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> ACTIVE(U33(mark(X)))
   MARK(U33(X)) -> U33^1(mark(X))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> U41^1(mark(X1), X2, X3)
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   MARK(U42(X1, X2, X3)) -> U42^1(mark(X1), X2, X3)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U43(X1, X2, X3)) -> U43^1(mark(X1), X2, X3)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   MARK(U44(X1, X2, X3)) -> U44^1(mark(X1), X2, X3)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U45(X1, X2)) -> U45^1(mark(X1), X2)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> ACTIVE(U46(mark(X)))
   MARK(U46(X)) -> U46^1(mark(X))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> U51^1(mark(X1), X2)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   MARK(U52(X)) -> U52^1(mark(X))
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> U61^1(mark(X1), X2)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   MARK(U62(X)) -> U62^1(mark(X))
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> ACTIVE(U71(mark(X)))
   MARK(U71(X)) -> U71^1(mark(X))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> ACTIVE(U81(mark(X)))
   MARK(U81(X)) -> U81^1(mark(X))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> U91^1(mark(X1), X2, X3)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> U92^1(mark(X1), X2, X3)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> U93^1(mark(X1), X2, X3)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   MARK(U94(X1, X2, X3)) -> U94^1(mark(X1), X2, X3)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   MARK(U95(X1, X2)) -> U95^1(mark(X1), X2)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> ACTIVE(U96(mark(X)))
   MARK(U96(X)) -> U96^1(mark(X))
   MARK(U96(X)) -> MARK(X)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(X1, mark(X2)) -> CONS(X1, X2)
   CONS(active(X1), X2) -> CONS(X1, X2)
   CONS(X1, active(X2)) -> CONS(X1, X2)
   U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3)
   U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3)
   U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3)
   U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3)
   ISNATKIND(mark(X)) -> ISNATKIND(X)
   ISNATKIND(active(X)) -> ISNATKIND(X)
   U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3)
   U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3)
   ISNATILISTKIND(mark(X)) -> ISNATILISTKIND(X)
   ISNATILISTKIND(active(X)) -> ISNATILISTKIND(X)
   U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3)
   U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3)
   U105^1(mark(X1), X2) -> U105^1(X1, X2)
   U105^1(X1, mark(X2)) -> U105^1(X1, X2)
   U105^1(active(X1), X2) -> U105^1(X1, X2)
   U105^1(X1, active(X2)) -> U105^1(X1, X2)
   ISNAT(mark(X)) -> ISNAT(X)
   ISNAT(active(X)) -> ISNAT(X)
   U106^1(mark(X)) -> U106^1(X)
   U106^1(active(X)) -> U106^1(X)
   ISNATILIST(mark(X)) -> ISNATILIST(X)
   ISNATILIST(active(X)) -> ISNATILIST(X)
   U11^1(mark(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
   U11^1(active(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, active(X2)) -> U11^1(X1, X2)
   U12^1(mark(X1), X2) -> U12^1(X1, X2)
   U12^1(X1, mark(X2)) -> U12^1(X1, X2)
   U12^1(active(X1), X2) -> U12^1(X1, X2)
   U12^1(X1, active(X2)) -> U12^1(X1, X2)
   U111^1(mark(X1), X2, X3) -> U111^1(X1, X2, X3)
   U111^1(X1, mark(X2), X3) -> U111^1(X1, X2, X3)
   U111^1(X1, X2, mark(X3)) -> U111^1(X1, X2, X3)
   U111^1(active(X1), X2, X3) -> U111^1(X1, X2, X3)
   U111^1(X1, active(X2), X3) -> U111^1(X1, X2, X3)
   U111^1(X1, X2, active(X3)) -> U111^1(X1, X2, X3)
   U112^1(mark(X1), X2, X3) -> U112^1(X1, X2, X3)
   U112^1(X1, mark(X2), X3) -> U112^1(X1, X2, X3)
   U112^1(X1, X2, mark(X3)) -> U112^1(X1, X2, X3)
   U112^1(active(X1), X2, X3) -> U112^1(X1, X2, X3)
   U112^1(X1, active(X2), X3) -> U112^1(X1, X2, X3)
   U112^1(X1, X2, active(X3)) -> U112^1(X1, X2, X3)
   U113^1(mark(X1), X2, X3) -> U113^1(X1, X2, X3)
   U113^1(X1, mark(X2), X3) -> U113^1(X1, X2, X3)
   U113^1(X1, X2, mark(X3)) -> U113^1(X1, X2, X3)
   U113^1(active(X1), X2, X3) -> U113^1(X1, X2, X3)
   U113^1(X1, active(X2), X3) -> U113^1(X1, X2, X3)
   U113^1(X1, X2, active(X3)) -> U113^1(X1, X2, X3)
   U114^1(mark(X1), X2) -> U114^1(X1, X2)
   U114^1(X1, mark(X2)) -> U114^1(X1, X2)
   U114^1(active(X1), X2) -> U114^1(X1, X2)
   U114^1(X1, active(X2)) -> U114^1(X1, X2)
   S(mark(X)) -> S(X)
   S(active(X)) -> S(X)
   LENGTH(mark(X)) -> LENGTH(X)
   LENGTH(active(X)) -> LENGTH(X)
   U13^1(mark(X)) -> U13^1(X)
   U13^1(active(X)) -> U13^1(X)
   ISNATLIST(mark(X)) -> ISNATLIST(X)
   ISNATLIST(active(X)) -> ISNATLIST(X)
   U121^1(mark(X1), X2) -> U121^1(X1, X2)
   U121^1(X1, mark(X2)) -> U121^1(X1, X2)
   U121^1(active(X1), X2) -> U121^1(X1, X2)
   U121^1(X1, active(X2)) -> U121^1(X1, X2)
   U122^1(mark(X)) -> U122^1(X)
   U122^1(active(X)) -> U122^1(X)
   U131^1(mark(X1), X2, X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, mark(X2), X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, mark(X3), X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, X3, mark(X4)) -> U131^1(X1, X2, X3, X4)
   U131^1(active(X1), X2, X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, active(X2), X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, active(X3), X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, X3, active(X4)) -> U131^1(X1, X2, X3, X4)
   U132^1(mark(X1), X2, X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, mark(X2), X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, mark(X3), X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, X3, mark(X4)) -> U132^1(X1, X2, X3, X4)
   U132^1(active(X1), X2, X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, active(X2), X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, active(X3), X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, X3, active(X4)) -> U132^1(X1, X2, X3, X4)
   U133^1(mark(X1), X2, X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, mark(X2), X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, mark(X3), X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, X3, mark(X4)) -> U133^1(X1, X2, X3, X4)
   U133^1(active(X1), X2, X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, active(X2), X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, active(X3), X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, X3, active(X4)) -> U133^1(X1, X2, X3, X4)
   U134^1(mark(X1), X2, X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, mark(X2), X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, mark(X3), X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, X3, mark(X4)) -> U134^1(X1, X2, X3, X4)
   U134^1(active(X1), X2, X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, active(X2), X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, active(X3), X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, X3, active(X4)) -> U134^1(X1, X2, X3, X4)
   U135^1(mark(X1), X2, X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, mark(X2), X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, mark(X3), X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, X3, mark(X4)) -> U135^1(X1, X2, X3, X4)
   U135^1(active(X1), X2, X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, active(X2), X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, active(X3), X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, X3, active(X4)) -> U135^1(X1, X2, X3, X4)
   U136^1(mark(X1), X2, X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, mark(X2), X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, mark(X3), X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, X3, mark(X4)) -> U136^1(X1, X2, X3, X4)
   U136^1(active(X1), X2, X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, active(X2), X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, active(X3), X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, X3, active(X4)) -> U136^1(X1, X2, X3, X4)
   TAKE(mark(X1), X2) -> TAKE(X1, X2)
   TAKE(X1, mark(X2)) -> TAKE(X1, X2)
   TAKE(active(X1), X2) -> TAKE(X1, X2)
   TAKE(X1, active(X2)) -> TAKE(X1, X2)
   U21^1(mark(X1), X2) -> U21^1(X1, X2)
   U21^1(X1, mark(X2)) -> U21^1(X1, X2)
   U21^1(active(X1), X2) -> U21^1(X1, X2)
   U21^1(X1, active(X2)) -> U21^1(X1, X2)
   U22^1(mark(X1), X2) -> U22^1(X1, X2)
   U22^1(X1, mark(X2)) -> U22^1(X1, X2)
   U22^1(active(X1), X2) -> U22^1(X1, X2)
   U22^1(X1, active(X2)) -> U22^1(X1, X2)
   U23^1(mark(X)) -> U23^1(X)
   U23^1(active(X)) -> U23^1(X)
   U31^1(mark(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
   U31^1(active(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, active(X2)) -> U31^1(X1, X2)
   U32^1(mark(X1), X2) -> U32^1(X1, X2)
   U32^1(X1, mark(X2)) -> U32^1(X1, X2)
   U32^1(active(X1), X2) -> U32^1(X1, X2)
   U32^1(X1, active(X2)) -> U32^1(X1, X2)
   U33^1(mark(X)) -> U33^1(X)
   U33^1(active(X)) -> U33^1(X)
   U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
   U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
   U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
   U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
   U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
   U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)
   U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
   U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
   U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
   U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
   U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
   U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)
   U43^1(mark(X1), X2, X3) -> U43^1(X1, X2, X3)
   U43^1(X1, mark(X2), X3) -> U43^1(X1, X2, X3)
   U43^1(X1, X2, mark(X3)) -> U43^1(X1, X2, X3)
   U43^1(active(X1), X2, X3) -> U43^1(X1, X2, X3)
   U43^1(X1, active(X2), X3) -> U43^1(X1, X2, X3)
   U43^1(X1, X2, active(X3)) -> U43^1(X1, X2, X3)
   U44^1(mark(X1), X2, X3) -> U44^1(X1, X2, X3)
   U44^1(X1, mark(X2), X3) -> U44^1(X1, X2, X3)
   U44^1(X1, X2, mark(X3)) -> U44^1(X1, X2, X3)
   U44^1(active(X1), X2, X3) -> U44^1(X1, X2, X3)
   U44^1(X1, active(X2), X3) -> U44^1(X1, X2, X3)
   U44^1(X1, X2, active(X3)) -> U44^1(X1, X2, X3)
   U45^1(mark(X1), X2) -> U45^1(X1, X2)
   U45^1(X1, mark(X2)) -> U45^1(X1, X2)
   U45^1(active(X1), X2) -> U45^1(X1, X2)
   U45^1(X1, active(X2)) -> U45^1(X1, X2)
   U46^1(mark(X)) -> U46^1(X)
   U46^1(active(X)) -> U46^1(X)
   U51^1(mark(X1), X2) -> U51^1(X1, X2)
   U51^1(X1, mark(X2)) -> U51^1(X1, X2)
   U51^1(active(X1), X2) -> U51^1(X1, X2)
   U51^1(X1, active(X2)) -> U51^1(X1, X2)
   U52^1(mark(X)) -> U52^1(X)
   U52^1(active(X)) -> U52^1(X)
   U61^1(mark(X1), X2) -> U61^1(X1, X2)
   U61^1(X1, mark(X2)) -> U61^1(X1, X2)
   U61^1(active(X1), X2) -> U61^1(X1, X2)
   U61^1(X1, active(X2)) -> U61^1(X1, X2)
   U62^1(mark(X)) -> U62^1(X)
   U62^1(active(X)) -> U62^1(X)
   U71^1(mark(X)) -> U71^1(X)
   U71^1(active(X)) -> U71^1(X)
   U81^1(mark(X)) -> U81^1(X)
   U81^1(active(X)) -> U81^1(X)
   U91^1(mark(X1), X2, X3) -> U91^1(X1, X2, X3)
   U91^1(X1, mark(X2), X3) -> U91^1(X1, X2, X3)
   U91^1(X1, X2, mark(X3)) -> U91^1(X1, X2, X3)
   U91^1(active(X1), X2, X3) -> U91^1(X1, X2, X3)
   U91^1(X1, active(X2), X3) -> U91^1(X1, X2, X3)
   U91^1(X1, X2, active(X3)) -> U91^1(X1, X2, X3)
   U92^1(mark(X1), X2, X3) -> U92^1(X1, X2, X3)
   U92^1(X1, mark(X2), X3) -> U92^1(X1, X2, X3)
   U92^1(X1, X2, mark(X3)) -> U92^1(X1, X2, X3)
   U92^1(active(X1), X2, X3) -> U92^1(X1, X2, X3)
   U92^1(X1, active(X2), X3) -> U92^1(X1, X2, X3)
   U92^1(X1, X2, active(X3)) -> U92^1(X1, X2, X3)
   U93^1(mark(X1), X2, X3) -> U93^1(X1, X2, X3)
   U93^1(X1, mark(X2), X3) -> U93^1(X1, X2, X3)
   U93^1(X1, X2, mark(X3)) -> U93^1(X1, X2, X3)
   U93^1(active(X1), X2, X3) -> U93^1(X1, X2, X3)
   U93^1(X1, active(X2), X3) -> U93^1(X1, X2, X3)
   U93^1(X1, X2, active(X3)) -> U93^1(X1, X2, X3)
   U94^1(mark(X1), X2, X3) -> U94^1(X1, X2, X3)
   U94^1(X1, mark(X2), X3) -> U94^1(X1, X2, X3)
   U94^1(X1, X2, mark(X3)) -> U94^1(X1, X2, X3)
   U94^1(active(X1), X2, X3) -> U94^1(X1, X2, X3)
   U94^1(X1, active(X2), X3) -> U94^1(X1, X2, X3)
   U94^1(X1, X2, active(X3)) -> U94^1(X1, X2, X3)
   U95^1(mark(X1), X2) -> U95^1(X1, X2)
   U95^1(X1, mark(X2)) -> U95^1(X1, X2)
   U95^1(active(X1), X2) -> U95^1(X1, X2)
   U95^1(X1, active(X2)) -> U95^1(X1, X2)
   U96^1(mark(X)) -> U96^1(X)
   U96^1(active(X)) -> U96^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 55 SCCs with 165 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

----------------------------------------

(5)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U96^1(active(X)) -> U96^1(X)
   U96^1(mark(X)) -> U96^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(6) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U96^1(active(X)) -> U96^1(X)
   U96^1(mark(X)) -> U96^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(8) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U96^1(active(X)) -> U96^1(X)
   U96^1(mark(X)) -> U96^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(10) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U96^1(active(X)) -> U96^1(X)
The graph contains the following edges 1 > 1


*U96^1(mark(X)) -> U96^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(11)
YES

----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U95^1(X1, mark(X2)) -> U95^1(X1, X2)
   U95^1(mark(X1), X2) -> U95^1(X1, X2)
   U95^1(active(X1), X2) -> U95^1(X1, X2)
   U95^1(X1, active(X2)) -> U95^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U95^1(X1, mark(X2)) -> U95^1(X1, X2)
   U95^1(mark(X1), X2) -> U95^1(X1, X2)
   U95^1(active(X1), X2) -> U95^1(X1, X2)
   U95^1(X1, active(X2)) -> U95^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(15) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U95^1(X1, mark(X2)) -> U95^1(X1, X2)
   U95^1(mark(X1), X2) -> U95^1(X1, X2)
   U95^1(active(X1), X2) -> U95^1(X1, X2)
   U95^1(X1, active(X2)) -> U95^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U95^1(X1, mark(X2)) -> U95^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U95^1(mark(X1), X2) -> U95^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U95^1(active(X1), X2) -> U95^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U95^1(X1, active(X2)) -> U95^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(18)
YES

----------------------------------------

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U94^1(X1, mark(X2), X3) -> U94^1(X1, X2, X3)
   U94^1(mark(X1), X2, X3) -> U94^1(X1, X2, X3)
   U94^1(X1, X2, mark(X3)) -> U94^1(X1, X2, X3)
   U94^1(active(X1), X2, X3) -> U94^1(X1, X2, X3)
   U94^1(X1, active(X2), X3) -> U94^1(X1, X2, X3)
   U94^1(X1, X2, active(X3)) -> U94^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(20) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U94^1(X1, mark(X2), X3) -> U94^1(X1, X2, X3)
   U94^1(mark(X1), X2, X3) -> U94^1(X1, X2, X3)
   U94^1(X1, X2, mark(X3)) -> U94^1(X1, X2, X3)
   U94^1(active(X1), X2, X3) -> U94^1(X1, X2, X3)
   U94^1(X1, active(X2), X3) -> U94^1(X1, X2, X3)
   U94^1(X1, X2, active(X3)) -> U94^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(22) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U94^1(X1, mark(X2), X3) -> U94^1(X1, X2, X3)
   U94^1(mark(X1), X2, X3) -> U94^1(X1, X2, X3)
   U94^1(X1, X2, mark(X3)) -> U94^1(X1, X2, X3)
   U94^1(active(X1), X2, X3) -> U94^1(X1, X2, X3)
   U94^1(X1, active(X2), X3) -> U94^1(X1, X2, X3)
   U94^1(X1, X2, active(X3)) -> U94^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(24) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U94^1(X1, mark(X2), X3) -> U94^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U94^1(mark(X1), X2, X3) -> U94^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U94^1(X1, X2, mark(X3)) -> U94^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U94^1(active(X1), X2, X3) -> U94^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U94^1(X1, active(X2), X3) -> U94^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U94^1(X1, X2, active(X3)) -> U94^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(25)
YES

----------------------------------------

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U93^1(X1, mark(X2), X3) -> U93^1(X1, X2, X3)
   U93^1(mark(X1), X2, X3) -> U93^1(X1, X2, X3)
   U93^1(X1, X2, mark(X3)) -> U93^1(X1, X2, X3)
   U93^1(active(X1), X2, X3) -> U93^1(X1, X2, X3)
   U93^1(X1, active(X2), X3) -> U93^1(X1, X2, X3)
   U93^1(X1, X2, active(X3)) -> U93^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(27) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U93^1(X1, mark(X2), X3) -> U93^1(X1, X2, X3)
   U93^1(mark(X1), X2, X3) -> U93^1(X1, X2, X3)
   U93^1(X1, X2, mark(X3)) -> U93^1(X1, X2, X3)
   U93^1(active(X1), X2, X3) -> U93^1(X1, X2, X3)
   U93^1(X1, active(X2), X3) -> U93^1(X1, X2, X3)
   U93^1(X1, X2, active(X3)) -> U93^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(29) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U93^1(X1, mark(X2), X3) -> U93^1(X1, X2, X3)
   U93^1(mark(X1), X2, X3) -> U93^1(X1, X2, X3)
   U93^1(X1, X2, mark(X3)) -> U93^1(X1, X2, X3)
   U93^1(active(X1), X2, X3) -> U93^1(X1, X2, X3)
   U93^1(X1, active(X2), X3) -> U93^1(X1, X2, X3)
   U93^1(X1, X2, active(X3)) -> U93^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(31) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U93^1(X1, mark(X2), X3) -> U93^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U93^1(mark(X1), X2, X3) -> U93^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U93^1(X1, X2, mark(X3)) -> U93^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U93^1(active(X1), X2, X3) -> U93^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U93^1(X1, active(X2), X3) -> U93^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U93^1(X1, X2, active(X3)) -> U93^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(32)
YES

----------------------------------------

(33)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U92^1(X1, mark(X2), X3) -> U92^1(X1, X2, X3)
   U92^1(mark(X1), X2, X3) -> U92^1(X1, X2, X3)
   U92^1(X1, X2, mark(X3)) -> U92^1(X1, X2, X3)
   U92^1(active(X1), X2, X3) -> U92^1(X1, X2, X3)
   U92^1(X1, active(X2), X3) -> U92^1(X1, X2, X3)
   U92^1(X1, X2, active(X3)) -> U92^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(34) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(35)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U92^1(X1, mark(X2), X3) -> U92^1(X1, X2, X3)
   U92^1(mark(X1), X2, X3) -> U92^1(X1, X2, X3)
   U92^1(X1, X2, mark(X3)) -> U92^1(X1, X2, X3)
   U92^1(active(X1), X2, X3) -> U92^1(X1, X2, X3)
   U92^1(X1, active(X2), X3) -> U92^1(X1, X2, X3)
   U92^1(X1, X2, active(X3)) -> U92^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(36) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(37)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U92^1(X1, mark(X2), X3) -> U92^1(X1, X2, X3)
   U92^1(mark(X1), X2, X3) -> U92^1(X1, X2, X3)
   U92^1(X1, X2, mark(X3)) -> U92^1(X1, X2, X3)
   U92^1(active(X1), X2, X3) -> U92^1(X1, X2, X3)
   U92^1(X1, active(X2), X3) -> U92^1(X1, X2, X3)
   U92^1(X1, X2, active(X3)) -> U92^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(38) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U92^1(X1, mark(X2), X3) -> U92^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U92^1(mark(X1), X2, X3) -> U92^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U92^1(X1, X2, mark(X3)) -> U92^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U92^1(active(X1), X2, X3) -> U92^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U92^1(X1, active(X2), X3) -> U92^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U92^1(X1, X2, active(X3)) -> U92^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(39)
YES

----------------------------------------

(40)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U91^1(X1, mark(X2), X3) -> U91^1(X1, X2, X3)
   U91^1(mark(X1), X2, X3) -> U91^1(X1, X2, X3)
   U91^1(X1, X2, mark(X3)) -> U91^1(X1, X2, X3)
   U91^1(active(X1), X2, X3) -> U91^1(X1, X2, X3)
   U91^1(X1, active(X2), X3) -> U91^1(X1, X2, X3)
   U91^1(X1, X2, active(X3)) -> U91^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(41) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(42)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U91^1(X1, mark(X2), X3) -> U91^1(X1, X2, X3)
   U91^1(mark(X1), X2, X3) -> U91^1(X1, X2, X3)
   U91^1(X1, X2, mark(X3)) -> U91^1(X1, X2, X3)
   U91^1(active(X1), X2, X3) -> U91^1(X1, X2, X3)
   U91^1(X1, active(X2), X3) -> U91^1(X1, X2, X3)
   U91^1(X1, X2, active(X3)) -> U91^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(43) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(44)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U91^1(X1, mark(X2), X3) -> U91^1(X1, X2, X3)
   U91^1(mark(X1), X2, X3) -> U91^1(X1, X2, X3)
   U91^1(X1, X2, mark(X3)) -> U91^1(X1, X2, X3)
   U91^1(active(X1), X2, X3) -> U91^1(X1, X2, X3)
   U91^1(X1, active(X2), X3) -> U91^1(X1, X2, X3)
   U91^1(X1, X2, active(X3)) -> U91^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(45) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U91^1(X1, mark(X2), X3) -> U91^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U91^1(mark(X1), X2, X3) -> U91^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U91^1(X1, X2, mark(X3)) -> U91^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U91^1(active(X1), X2, X3) -> U91^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U91^1(X1, active(X2), X3) -> U91^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U91^1(X1, X2, active(X3)) -> U91^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(46)
YES

----------------------------------------

(47)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U81^1(active(X)) -> U81^1(X)
   U81^1(mark(X)) -> U81^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(48) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(49)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U81^1(active(X)) -> U81^1(X)
   U81^1(mark(X)) -> U81^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(50) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(51)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U81^1(active(X)) -> U81^1(X)
   U81^1(mark(X)) -> U81^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(52) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U81^1(active(X)) -> U81^1(X)
The graph contains the following edges 1 > 1


*U81^1(mark(X)) -> U81^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(53)
YES

----------------------------------------

(54)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U71^1(active(X)) -> U71^1(X)
   U71^1(mark(X)) -> U71^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(55) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(56)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U71^1(active(X)) -> U71^1(X)
   U71^1(mark(X)) -> U71^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(57) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(58)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U71^1(active(X)) -> U71^1(X)
   U71^1(mark(X)) -> U71^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(59) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U71^1(active(X)) -> U71^1(X)
The graph contains the following edges 1 > 1


*U71^1(mark(X)) -> U71^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(60)
YES

----------------------------------------

(61)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U62^1(active(X)) -> U62^1(X)
   U62^1(mark(X)) -> U62^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(62) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(63)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U62^1(active(X)) -> U62^1(X)
   U62^1(mark(X)) -> U62^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(64) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(65)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U62^1(active(X)) -> U62^1(X)
   U62^1(mark(X)) -> U62^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(66) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U62^1(active(X)) -> U62^1(X)
The graph contains the following edges 1 > 1


*U62^1(mark(X)) -> U62^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(67)
YES

----------------------------------------

(68)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U61^1(X1, mark(X2)) -> U61^1(X1, X2)
   U61^1(mark(X1), X2) -> U61^1(X1, X2)
   U61^1(active(X1), X2) -> U61^1(X1, X2)
   U61^1(X1, active(X2)) -> U61^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(69) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(70)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U61^1(X1, mark(X2)) -> U61^1(X1, X2)
   U61^1(mark(X1), X2) -> U61^1(X1, X2)
   U61^1(active(X1), X2) -> U61^1(X1, X2)
   U61^1(X1, active(X2)) -> U61^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(71) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(72)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U61^1(X1, mark(X2)) -> U61^1(X1, X2)
   U61^1(mark(X1), X2) -> U61^1(X1, X2)
   U61^1(active(X1), X2) -> U61^1(X1, X2)
   U61^1(X1, active(X2)) -> U61^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(73) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U61^1(X1, mark(X2)) -> U61^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U61^1(mark(X1), X2) -> U61^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U61^1(active(X1), X2) -> U61^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U61^1(X1, active(X2)) -> U61^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(74)
YES

----------------------------------------

(75)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U52^1(active(X)) -> U52^1(X)
   U52^1(mark(X)) -> U52^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(76) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(77)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U52^1(active(X)) -> U52^1(X)
   U52^1(mark(X)) -> U52^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(78) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(79)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U52^1(active(X)) -> U52^1(X)
   U52^1(mark(X)) -> U52^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(80) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U52^1(active(X)) -> U52^1(X)
The graph contains the following edges 1 > 1


*U52^1(mark(X)) -> U52^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(81)
YES

----------------------------------------

(82)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(X1, mark(X2)) -> U51^1(X1, X2)
   U51^1(mark(X1), X2) -> U51^1(X1, X2)
   U51^1(active(X1), X2) -> U51^1(X1, X2)
   U51^1(X1, active(X2)) -> U51^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(83) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(84)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(X1, mark(X2)) -> U51^1(X1, X2)
   U51^1(mark(X1), X2) -> U51^1(X1, X2)
   U51^1(active(X1), X2) -> U51^1(X1, X2)
   U51^1(X1, active(X2)) -> U51^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(85) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(86)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U51^1(X1, mark(X2)) -> U51^1(X1, X2)
   U51^1(mark(X1), X2) -> U51^1(X1, X2)
   U51^1(active(X1), X2) -> U51^1(X1, X2)
   U51^1(X1, active(X2)) -> U51^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(87) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U51^1(X1, mark(X2)) -> U51^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U51^1(mark(X1), X2) -> U51^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U51^1(active(X1), X2) -> U51^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U51^1(X1, active(X2)) -> U51^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(88)
YES

----------------------------------------

(89)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U46^1(active(X)) -> U46^1(X)
   U46^1(mark(X)) -> U46^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(90) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(91)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U46^1(active(X)) -> U46^1(X)
   U46^1(mark(X)) -> U46^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(92) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(93)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U46^1(active(X)) -> U46^1(X)
   U46^1(mark(X)) -> U46^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(94) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U46^1(active(X)) -> U46^1(X)
The graph contains the following edges 1 > 1


*U46^1(mark(X)) -> U46^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(95)
YES

----------------------------------------

(96)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U45^1(X1, mark(X2)) -> U45^1(X1, X2)
   U45^1(mark(X1), X2) -> U45^1(X1, X2)
   U45^1(active(X1), X2) -> U45^1(X1, X2)
   U45^1(X1, active(X2)) -> U45^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(97) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(98)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U45^1(X1, mark(X2)) -> U45^1(X1, X2)
   U45^1(mark(X1), X2) -> U45^1(X1, X2)
   U45^1(active(X1), X2) -> U45^1(X1, X2)
   U45^1(X1, active(X2)) -> U45^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(99) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(100)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U45^1(X1, mark(X2)) -> U45^1(X1, X2)
   U45^1(mark(X1), X2) -> U45^1(X1, X2)
   U45^1(active(X1), X2) -> U45^1(X1, X2)
   U45^1(X1, active(X2)) -> U45^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(101) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U45^1(X1, mark(X2)) -> U45^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U45^1(mark(X1), X2) -> U45^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U45^1(active(X1), X2) -> U45^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U45^1(X1, active(X2)) -> U45^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(102)
YES

----------------------------------------

(103)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U44^1(X1, mark(X2), X3) -> U44^1(X1, X2, X3)
   U44^1(mark(X1), X2, X3) -> U44^1(X1, X2, X3)
   U44^1(X1, X2, mark(X3)) -> U44^1(X1, X2, X3)
   U44^1(active(X1), X2, X3) -> U44^1(X1, X2, X3)
   U44^1(X1, active(X2), X3) -> U44^1(X1, X2, X3)
   U44^1(X1, X2, active(X3)) -> U44^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(104) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(105)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U44^1(X1, mark(X2), X3) -> U44^1(X1, X2, X3)
   U44^1(mark(X1), X2, X3) -> U44^1(X1, X2, X3)
   U44^1(X1, X2, mark(X3)) -> U44^1(X1, X2, X3)
   U44^1(active(X1), X2, X3) -> U44^1(X1, X2, X3)
   U44^1(X1, active(X2), X3) -> U44^1(X1, X2, X3)
   U44^1(X1, X2, active(X3)) -> U44^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(106) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(107)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U44^1(X1, mark(X2), X3) -> U44^1(X1, X2, X3)
   U44^1(mark(X1), X2, X3) -> U44^1(X1, X2, X3)
   U44^1(X1, X2, mark(X3)) -> U44^1(X1, X2, X3)
   U44^1(active(X1), X2, X3) -> U44^1(X1, X2, X3)
   U44^1(X1, active(X2), X3) -> U44^1(X1, X2, X3)
   U44^1(X1, X2, active(X3)) -> U44^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(108) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U44^1(X1, mark(X2), X3) -> U44^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U44^1(mark(X1), X2, X3) -> U44^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U44^1(X1, X2, mark(X3)) -> U44^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U44^1(active(X1), X2, X3) -> U44^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U44^1(X1, active(X2), X3) -> U44^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U44^1(X1, X2, active(X3)) -> U44^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(109)
YES

----------------------------------------

(110)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U43^1(X1, mark(X2), X3) -> U43^1(X1, X2, X3)
   U43^1(mark(X1), X2, X3) -> U43^1(X1, X2, X3)
   U43^1(X1, X2, mark(X3)) -> U43^1(X1, X2, X3)
   U43^1(active(X1), X2, X3) -> U43^1(X1, X2, X3)
   U43^1(X1, active(X2), X3) -> U43^1(X1, X2, X3)
   U43^1(X1, X2, active(X3)) -> U43^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(111) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(112)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U43^1(X1, mark(X2), X3) -> U43^1(X1, X2, X3)
   U43^1(mark(X1), X2, X3) -> U43^1(X1, X2, X3)
   U43^1(X1, X2, mark(X3)) -> U43^1(X1, X2, X3)
   U43^1(active(X1), X2, X3) -> U43^1(X1, X2, X3)
   U43^1(X1, active(X2), X3) -> U43^1(X1, X2, X3)
   U43^1(X1, X2, active(X3)) -> U43^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(113) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(114)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U43^1(X1, mark(X2), X3) -> U43^1(X1, X2, X3)
   U43^1(mark(X1), X2, X3) -> U43^1(X1, X2, X3)
   U43^1(X1, X2, mark(X3)) -> U43^1(X1, X2, X3)
   U43^1(active(X1), X2, X3) -> U43^1(X1, X2, X3)
   U43^1(X1, active(X2), X3) -> U43^1(X1, X2, X3)
   U43^1(X1, X2, active(X3)) -> U43^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(115) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U43^1(X1, mark(X2), X3) -> U43^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U43^1(mark(X1), X2, X3) -> U43^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U43^1(X1, X2, mark(X3)) -> U43^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U43^1(active(X1), X2, X3) -> U43^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U43^1(X1, active(X2), X3) -> U43^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U43^1(X1, X2, active(X3)) -> U43^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(116)
YES

----------------------------------------

(117)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
   U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
   U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
   U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
   U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
   U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(118) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(119)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
   U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
   U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
   U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
   U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
   U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(120) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(121)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
   U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
   U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
   U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
   U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
   U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(122) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U42^1(X1, mark(X2), X3) -> U42^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U42^1(mark(X1), X2, X3) -> U42^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U42^1(X1, X2, mark(X3)) -> U42^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U42^1(active(X1), X2, X3) -> U42^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U42^1(X1, active(X2), X3) -> U42^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U42^1(X1, X2, active(X3)) -> U42^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(123)
YES

----------------------------------------

(124)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
   U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
   U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
   U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
   U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
   U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(125) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(126)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
   U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
   U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
   U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
   U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
   U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(127) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(128)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
   U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
   U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
   U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
   U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
   U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(129) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U41^1(X1, mark(X2), X3) -> U41^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U41^1(mark(X1), X2, X3) -> U41^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U41^1(X1, X2, mark(X3)) -> U41^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U41^1(active(X1), X2, X3) -> U41^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U41^1(X1, active(X2), X3) -> U41^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U41^1(X1, X2, active(X3)) -> U41^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(130)
YES

----------------------------------------

(131)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U33^1(active(X)) -> U33^1(X)
   U33^1(mark(X)) -> U33^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(132) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(133)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U33^1(active(X)) -> U33^1(X)
   U33^1(mark(X)) -> U33^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(134) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(135)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U33^1(active(X)) -> U33^1(X)
   U33^1(mark(X)) -> U33^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(136) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U33^1(active(X)) -> U33^1(X)
The graph contains the following edges 1 > 1


*U33^1(mark(X)) -> U33^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(137)
YES

----------------------------------------

(138)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U32^1(X1, mark(X2)) -> U32^1(X1, X2)
   U32^1(mark(X1), X2) -> U32^1(X1, X2)
   U32^1(active(X1), X2) -> U32^1(X1, X2)
   U32^1(X1, active(X2)) -> U32^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(139) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(140)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U32^1(X1, mark(X2)) -> U32^1(X1, X2)
   U32^1(mark(X1), X2) -> U32^1(X1, X2)
   U32^1(active(X1), X2) -> U32^1(X1, X2)
   U32^1(X1, active(X2)) -> U32^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(141) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(142)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U32^1(X1, mark(X2)) -> U32^1(X1, X2)
   U32^1(mark(X1), X2) -> U32^1(X1, X2)
   U32^1(active(X1), X2) -> U32^1(X1, X2)
   U32^1(X1, active(X2)) -> U32^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(143) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U32^1(X1, mark(X2)) -> U32^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U32^1(mark(X1), X2) -> U32^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U32^1(active(X1), X2) -> U32^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U32^1(X1, active(X2)) -> U32^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(144)
YES

----------------------------------------

(145)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
   U31^1(mark(X1), X2) -> U31^1(X1, X2)
   U31^1(active(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, active(X2)) -> U31^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(146) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(147)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
   U31^1(mark(X1), X2) -> U31^1(X1, X2)
   U31^1(active(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, active(X2)) -> U31^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(148) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(149)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U31^1(X1, mark(X2)) -> U31^1(X1, X2)
   U31^1(mark(X1), X2) -> U31^1(X1, X2)
   U31^1(active(X1), X2) -> U31^1(X1, X2)
   U31^1(X1, active(X2)) -> U31^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(150) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U31^1(X1, mark(X2)) -> U31^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U31^1(mark(X1), X2) -> U31^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U31^1(active(X1), X2) -> U31^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U31^1(X1, active(X2)) -> U31^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(151)
YES

----------------------------------------

(152)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U23^1(active(X)) -> U23^1(X)
   U23^1(mark(X)) -> U23^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(153) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(154)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U23^1(active(X)) -> U23^1(X)
   U23^1(mark(X)) -> U23^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(155) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(156)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U23^1(active(X)) -> U23^1(X)
   U23^1(mark(X)) -> U23^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(157) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U23^1(active(X)) -> U23^1(X)
The graph contains the following edges 1 > 1


*U23^1(mark(X)) -> U23^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(158)
YES

----------------------------------------

(159)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U22^1(X1, mark(X2)) -> U22^1(X1, X2)
   U22^1(mark(X1), X2) -> U22^1(X1, X2)
   U22^1(active(X1), X2) -> U22^1(X1, X2)
   U22^1(X1, active(X2)) -> U22^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(160) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(161)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U22^1(X1, mark(X2)) -> U22^1(X1, X2)
   U22^1(mark(X1), X2) -> U22^1(X1, X2)
   U22^1(active(X1), X2) -> U22^1(X1, X2)
   U22^1(X1, active(X2)) -> U22^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(162) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(163)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U22^1(X1, mark(X2)) -> U22^1(X1, X2)
   U22^1(mark(X1), X2) -> U22^1(X1, X2)
   U22^1(active(X1), X2) -> U22^1(X1, X2)
   U22^1(X1, active(X2)) -> U22^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(164) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U22^1(X1, mark(X2)) -> U22^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U22^1(mark(X1), X2) -> U22^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U22^1(active(X1), X2) -> U22^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U22^1(X1, active(X2)) -> U22^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(165)
YES

----------------------------------------

(166)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U21^1(X1, mark(X2)) -> U21^1(X1, X2)
   U21^1(mark(X1), X2) -> U21^1(X1, X2)
   U21^1(active(X1), X2) -> U21^1(X1, X2)
   U21^1(X1, active(X2)) -> U21^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(167) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(168)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U21^1(X1, mark(X2)) -> U21^1(X1, X2)
   U21^1(mark(X1), X2) -> U21^1(X1, X2)
   U21^1(active(X1), X2) -> U21^1(X1, X2)
   U21^1(X1, active(X2)) -> U21^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(169) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(170)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U21^1(X1, mark(X2)) -> U21^1(X1, X2)
   U21^1(mark(X1), X2) -> U21^1(X1, X2)
   U21^1(active(X1), X2) -> U21^1(X1, X2)
   U21^1(X1, active(X2)) -> U21^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(171) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U21^1(X1, mark(X2)) -> U21^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U21^1(mark(X1), X2) -> U21^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U21^1(active(X1), X2) -> U21^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U21^1(X1, active(X2)) -> U21^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(172)
YES

----------------------------------------

(173)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TAKE(X1, mark(X2)) -> TAKE(X1, X2)
   TAKE(mark(X1), X2) -> TAKE(X1, X2)
   TAKE(active(X1), X2) -> TAKE(X1, X2)
   TAKE(X1, active(X2)) -> TAKE(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(174) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(175)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TAKE(X1, mark(X2)) -> TAKE(X1, X2)
   TAKE(mark(X1), X2) -> TAKE(X1, X2)
   TAKE(active(X1), X2) -> TAKE(X1, X2)
   TAKE(X1, active(X2)) -> TAKE(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(176) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(177)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TAKE(X1, mark(X2)) -> TAKE(X1, X2)
   TAKE(mark(X1), X2) -> TAKE(X1, X2)
   TAKE(active(X1), X2) -> TAKE(X1, X2)
   TAKE(X1, active(X2)) -> TAKE(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(178) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*TAKE(X1, mark(X2)) -> TAKE(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*TAKE(mark(X1), X2) -> TAKE(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*TAKE(active(X1), X2) -> TAKE(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*TAKE(X1, active(X2)) -> TAKE(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(179)
YES

----------------------------------------

(180)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U136^1(X1, mark(X2), X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(mark(X1), X2, X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, mark(X3), X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, X3, mark(X4)) -> U136^1(X1, X2, X3, X4)
   U136^1(active(X1), X2, X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, active(X2), X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, active(X3), X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, X3, active(X4)) -> U136^1(X1, X2, X3, X4)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(181) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(182)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U136^1(X1, mark(X2), X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(mark(X1), X2, X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, mark(X3), X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, X3, mark(X4)) -> U136^1(X1, X2, X3, X4)
   U136^1(active(X1), X2, X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, active(X2), X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, active(X3), X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, X3, active(X4)) -> U136^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(183) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(184)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U136^1(X1, mark(X2), X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(mark(X1), X2, X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, mark(X3), X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, X3, mark(X4)) -> U136^1(X1, X2, X3, X4)
   U136^1(active(X1), X2, X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, active(X2), X3, X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, active(X3), X4) -> U136^1(X1, X2, X3, X4)
   U136^1(X1, X2, X3, active(X4)) -> U136^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(185) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U136^1(X1, mark(X2), X3, X4) -> U136^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U136^1(mark(X1), X2, X3, X4) -> U136^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U136^1(X1, X2, mark(X3), X4) -> U136^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U136^1(X1, X2, X3, mark(X4)) -> U136^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


*U136^1(active(X1), X2, X3, X4) -> U136^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U136^1(X1, active(X2), X3, X4) -> U136^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U136^1(X1, X2, active(X3), X4) -> U136^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U136^1(X1, X2, X3, active(X4)) -> U136^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


----------------------------------------

(186)
YES

----------------------------------------

(187)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U135^1(X1, mark(X2), X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(mark(X1), X2, X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, mark(X3), X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, X3, mark(X4)) -> U135^1(X1, X2, X3, X4)
   U135^1(active(X1), X2, X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, active(X2), X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, active(X3), X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, X3, active(X4)) -> U135^1(X1, X2, X3, X4)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(188) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(189)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U135^1(X1, mark(X2), X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(mark(X1), X2, X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, mark(X3), X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, X3, mark(X4)) -> U135^1(X1, X2, X3, X4)
   U135^1(active(X1), X2, X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, active(X2), X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, active(X3), X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, X3, active(X4)) -> U135^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(190) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(191)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U135^1(X1, mark(X2), X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(mark(X1), X2, X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, mark(X3), X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, X3, mark(X4)) -> U135^1(X1, X2, X3, X4)
   U135^1(active(X1), X2, X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, active(X2), X3, X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, active(X3), X4) -> U135^1(X1, X2, X3, X4)
   U135^1(X1, X2, X3, active(X4)) -> U135^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(192) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U135^1(X1, mark(X2), X3, X4) -> U135^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U135^1(mark(X1), X2, X3, X4) -> U135^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U135^1(X1, X2, mark(X3), X4) -> U135^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U135^1(X1, X2, X3, mark(X4)) -> U135^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


*U135^1(active(X1), X2, X3, X4) -> U135^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U135^1(X1, active(X2), X3, X4) -> U135^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U135^1(X1, X2, active(X3), X4) -> U135^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U135^1(X1, X2, X3, active(X4)) -> U135^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


----------------------------------------

(193)
YES

----------------------------------------

(194)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U134^1(X1, mark(X2), X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(mark(X1), X2, X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, mark(X3), X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, X3, mark(X4)) -> U134^1(X1, X2, X3, X4)
   U134^1(active(X1), X2, X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, active(X2), X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, active(X3), X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, X3, active(X4)) -> U134^1(X1, X2, X3, X4)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(195) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(196)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U134^1(X1, mark(X2), X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(mark(X1), X2, X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, mark(X3), X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, X3, mark(X4)) -> U134^1(X1, X2, X3, X4)
   U134^1(active(X1), X2, X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, active(X2), X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, active(X3), X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, X3, active(X4)) -> U134^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(197) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(198)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U134^1(X1, mark(X2), X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(mark(X1), X2, X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, mark(X3), X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, X3, mark(X4)) -> U134^1(X1, X2, X3, X4)
   U134^1(active(X1), X2, X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, active(X2), X3, X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, active(X3), X4) -> U134^1(X1, X2, X3, X4)
   U134^1(X1, X2, X3, active(X4)) -> U134^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(199) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U134^1(X1, mark(X2), X3, X4) -> U134^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U134^1(mark(X1), X2, X3, X4) -> U134^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U134^1(X1, X2, mark(X3), X4) -> U134^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U134^1(X1, X2, X3, mark(X4)) -> U134^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


*U134^1(active(X1), X2, X3, X4) -> U134^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U134^1(X1, active(X2), X3, X4) -> U134^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U134^1(X1, X2, active(X3), X4) -> U134^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U134^1(X1, X2, X3, active(X4)) -> U134^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


----------------------------------------

(200)
YES

----------------------------------------

(201)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U133^1(X1, mark(X2), X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(mark(X1), X2, X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, mark(X3), X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, X3, mark(X4)) -> U133^1(X1, X2, X3, X4)
   U133^1(active(X1), X2, X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, active(X2), X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, active(X3), X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, X3, active(X4)) -> U133^1(X1, X2, X3, X4)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(202) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(203)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U133^1(X1, mark(X2), X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(mark(X1), X2, X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, mark(X3), X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, X3, mark(X4)) -> U133^1(X1, X2, X3, X4)
   U133^1(active(X1), X2, X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, active(X2), X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, active(X3), X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, X3, active(X4)) -> U133^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(204) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(205)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U133^1(X1, mark(X2), X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(mark(X1), X2, X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, mark(X3), X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, X3, mark(X4)) -> U133^1(X1, X2, X3, X4)
   U133^1(active(X1), X2, X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, active(X2), X3, X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, active(X3), X4) -> U133^1(X1, X2, X3, X4)
   U133^1(X1, X2, X3, active(X4)) -> U133^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(206) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U133^1(X1, mark(X2), X3, X4) -> U133^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U133^1(mark(X1), X2, X3, X4) -> U133^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U133^1(X1, X2, mark(X3), X4) -> U133^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U133^1(X1, X2, X3, mark(X4)) -> U133^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


*U133^1(active(X1), X2, X3, X4) -> U133^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U133^1(X1, active(X2), X3, X4) -> U133^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U133^1(X1, X2, active(X3), X4) -> U133^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U133^1(X1, X2, X3, active(X4)) -> U133^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


----------------------------------------

(207)
YES

----------------------------------------

(208)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U132^1(X1, mark(X2), X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(mark(X1), X2, X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, mark(X3), X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, X3, mark(X4)) -> U132^1(X1, X2, X3, X4)
   U132^1(active(X1), X2, X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, active(X2), X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, active(X3), X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, X3, active(X4)) -> U132^1(X1, X2, X3, X4)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(209) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(210)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U132^1(X1, mark(X2), X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(mark(X1), X2, X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, mark(X3), X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, X3, mark(X4)) -> U132^1(X1, X2, X3, X4)
   U132^1(active(X1), X2, X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, active(X2), X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, active(X3), X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, X3, active(X4)) -> U132^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(211) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(212)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U132^1(X1, mark(X2), X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(mark(X1), X2, X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, mark(X3), X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, X3, mark(X4)) -> U132^1(X1, X2, X3, X4)
   U132^1(active(X1), X2, X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, active(X2), X3, X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, active(X3), X4) -> U132^1(X1, X2, X3, X4)
   U132^1(X1, X2, X3, active(X4)) -> U132^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(213) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U132^1(X1, mark(X2), X3, X4) -> U132^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U132^1(mark(X1), X2, X3, X4) -> U132^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U132^1(X1, X2, mark(X3), X4) -> U132^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U132^1(X1, X2, X3, mark(X4)) -> U132^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


*U132^1(active(X1), X2, X3, X4) -> U132^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U132^1(X1, active(X2), X3, X4) -> U132^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U132^1(X1, X2, active(X3), X4) -> U132^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U132^1(X1, X2, X3, active(X4)) -> U132^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


----------------------------------------

(214)
YES

----------------------------------------

(215)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U131^1(X1, mark(X2), X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(mark(X1), X2, X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, mark(X3), X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, X3, mark(X4)) -> U131^1(X1, X2, X3, X4)
   U131^1(active(X1), X2, X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, active(X2), X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, active(X3), X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, X3, active(X4)) -> U131^1(X1, X2, X3, X4)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(216) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(217)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U131^1(X1, mark(X2), X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(mark(X1), X2, X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, mark(X3), X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, X3, mark(X4)) -> U131^1(X1, X2, X3, X4)
   U131^1(active(X1), X2, X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, active(X2), X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, active(X3), X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, X3, active(X4)) -> U131^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(218) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(219)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U131^1(X1, mark(X2), X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(mark(X1), X2, X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, mark(X3), X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, X3, mark(X4)) -> U131^1(X1, X2, X3, X4)
   U131^1(active(X1), X2, X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, active(X2), X3, X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, active(X3), X4) -> U131^1(X1, X2, X3, X4)
   U131^1(X1, X2, X3, active(X4)) -> U131^1(X1, X2, X3, X4)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(220) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U131^1(X1, mark(X2), X3, X4) -> U131^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U131^1(mark(X1), X2, X3, X4) -> U131^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U131^1(X1, X2, mark(X3), X4) -> U131^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U131^1(X1, X2, X3, mark(X4)) -> U131^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


*U131^1(active(X1), X2, X3, X4) -> U131^1(X1, X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*U131^1(X1, active(X2), X3, X4) -> U131^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*U131^1(X1, X2, active(X3), X4) -> U131^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4


*U131^1(X1, X2, X3, active(X4)) -> U131^1(X1, X2, X3, X4)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4


----------------------------------------

(221)
YES

----------------------------------------

(222)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U122^1(active(X)) -> U122^1(X)
   U122^1(mark(X)) -> U122^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(223) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(224)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U122^1(active(X)) -> U122^1(X)
   U122^1(mark(X)) -> U122^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(225) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(226)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U122^1(active(X)) -> U122^1(X)
   U122^1(mark(X)) -> U122^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(227) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U122^1(active(X)) -> U122^1(X)
The graph contains the following edges 1 > 1


*U122^1(mark(X)) -> U122^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(228)
YES

----------------------------------------

(229)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U121^1(X1, mark(X2)) -> U121^1(X1, X2)
   U121^1(mark(X1), X2) -> U121^1(X1, X2)
   U121^1(active(X1), X2) -> U121^1(X1, X2)
   U121^1(X1, active(X2)) -> U121^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(230) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(231)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U121^1(X1, mark(X2)) -> U121^1(X1, X2)
   U121^1(mark(X1), X2) -> U121^1(X1, X2)
   U121^1(active(X1), X2) -> U121^1(X1, X2)
   U121^1(X1, active(X2)) -> U121^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(232) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(233)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U121^1(X1, mark(X2)) -> U121^1(X1, X2)
   U121^1(mark(X1), X2) -> U121^1(X1, X2)
   U121^1(active(X1), X2) -> U121^1(X1, X2)
   U121^1(X1, active(X2)) -> U121^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(234) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U121^1(X1, mark(X2)) -> U121^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U121^1(mark(X1), X2) -> U121^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U121^1(active(X1), X2) -> U121^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U121^1(X1, active(X2)) -> U121^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(235)
YES

----------------------------------------

(236)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(active(X)) -> ISNATLIST(X)
   ISNATLIST(mark(X)) -> ISNATLIST(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(237) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(238)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(active(X)) -> ISNATLIST(X)
   ISNATLIST(mark(X)) -> ISNATLIST(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(239) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(240)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATLIST(active(X)) -> ISNATLIST(X)
   ISNATLIST(mark(X)) -> ISNATLIST(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(241) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*ISNATLIST(active(X)) -> ISNATLIST(X)
The graph contains the following edges 1 > 1


*ISNATLIST(mark(X)) -> ISNATLIST(X)
The graph contains the following edges 1 > 1


----------------------------------------

(242)
YES

----------------------------------------

(243)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U13^1(active(X)) -> U13^1(X)
   U13^1(mark(X)) -> U13^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(244) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(245)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U13^1(active(X)) -> U13^1(X)
   U13^1(mark(X)) -> U13^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(246) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(247)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U13^1(active(X)) -> U13^1(X)
   U13^1(mark(X)) -> U13^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(248) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U13^1(active(X)) -> U13^1(X)
The graph contains the following edges 1 > 1


*U13^1(mark(X)) -> U13^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(249)
YES

----------------------------------------

(250)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LENGTH(active(X)) -> LENGTH(X)
   LENGTH(mark(X)) -> LENGTH(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(251) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(252)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LENGTH(active(X)) -> LENGTH(X)
   LENGTH(mark(X)) -> LENGTH(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(253) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(254)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LENGTH(active(X)) -> LENGTH(X)
   LENGTH(mark(X)) -> LENGTH(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(255) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*LENGTH(active(X)) -> LENGTH(X)
The graph contains the following edges 1 > 1


*LENGTH(mark(X)) -> LENGTH(X)
The graph contains the following edges 1 > 1


----------------------------------------

(256)
YES

----------------------------------------

(257)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(active(X)) -> S(X)
   S(mark(X)) -> S(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(258) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(259)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(active(X)) -> S(X)
   S(mark(X)) -> S(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(260) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(261)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(active(X)) -> S(X)
   S(mark(X)) -> S(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(262) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*S(active(X)) -> S(X)
The graph contains the following edges 1 > 1


*S(mark(X)) -> S(X)
The graph contains the following edges 1 > 1


----------------------------------------

(263)
YES

----------------------------------------

(264)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U114^1(X1, mark(X2)) -> U114^1(X1, X2)
   U114^1(mark(X1), X2) -> U114^1(X1, X2)
   U114^1(active(X1), X2) -> U114^1(X1, X2)
   U114^1(X1, active(X2)) -> U114^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(265) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(266)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U114^1(X1, mark(X2)) -> U114^1(X1, X2)
   U114^1(mark(X1), X2) -> U114^1(X1, X2)
   U114^1(active(X1), X2) -> U114^1(X1, X2)
   U114^1(X1, active(X2)) -> U114^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(267) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(268)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U114^1(X1, mark(X2)) -> U114^1(X1, X2)
   U114^1(mark(X1), X2) -> U114^1(X1, X2)
   U114^1(active(X1), X2) -> U114^1(X1, X2)
   U114^1(X1, active(X2)) -> U114^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(269) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U114^1(X1, mark(X2)) -> U114^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U114^1(mark(X1), X2) -> U114^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U114^1(active(X1), X2) -> U114^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U114^1(X1, active(X2)) -> U114^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(270)
YES

----------------------------------------

(271)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U113^1(X1, mark(X2), X3) -> U113^1(X1, X2, X3)
   U113^1(mark(X1), X2, X3) -> U113^1(X1, X2, X3)
   U113^1(X1, X2, mark(X3)) -> U113^1(X1, X2, X3)
   U113^1(active(X1), X2, X3) -> U113^1(X1, X2, X3)
   U113^1(X1, active(X2), X3) -> U113^1(X1, X2, X3)
   U113^1(X1, X2, active(X3)) -> U113^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(272) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(273)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U113^1(X1, mark(X2), X3) -> U113^1(X1, X2, X3)
   U113^1(mark(X1), X2, X3) -> U113^1(X1, X2, X3)
   U113^1(X1, X2, mark(X3)) -> U113^1(X1, X2, X3)
   U113^1(active(X1), X2, X3) -> U113^1(X1, X2, X3)
   U113^1(X1, active(X2), X3) -> U113^1(X1, X2, X3)
   U113^1(X1, X2, active(X3)) -> U113^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(274) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(275)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U113^1(X1, mark(X2), X3) -> U113^1(X1, X2, X3)
   U113^1(mark(X1), X2, X3) -> U113^1(X1, X2, X3)
   U113^1(X1, X2, mark(X3)) -> U113^1(X1, X2, X3)
   U113^1(active(X1), X2, X3) -> U113^1(X1, X2, X3)
   U113^1(X1, active(X2), X3) -> U113^1(X1, X2, X3)
   U113^1(X1, X2, active(X3)) -> U113^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(276) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U113^1(X1, mark(X2), X3) -> U113^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U113^1(mark(X1), X2, X3) -> U113^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U113^1(X1, X2, mark(X3)) -> U113^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U113^1(active(X1), X2, X3) -> U113^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U113^1(X1, active(X2), X3) -> U113^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U113^1(X1, X2, active(X3)) -> U113^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(277)
YES

----------------------------------------

(278)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U112^1(X1, mark(X2), X3) -> U112^1(X1, X2, X3)
   U112^1(mark(X1), X2, X3) -> U112^1(X1, X2, X3)
   U112^1(X1, X2, mark(X3)) -> U112^1(X1, X2, X3)
   U112^1(active(X1), X2, X3) -> U112^1(X1, X2, X3)
   U112^1(X1, active(X2), X3) -> U112^1(X1, X2, X3)
   U112^1(X1, X2, active(X3)) -> U112^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(279) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(280)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U112^1(X1, mark(X2), X3) -> U112^1(X1, X2, X3)
   U112^1(mark(X1), X2, X3) -> U112^1(X1, X2, X3)
   U112^1(X1, X2, mark(X3)) -> U112^1(X1, X2, X3)
   U112^1(active(X1), X2, X3) -> U112^1(X1, X2, X3)
   U112^1(X1, active(X2), X3) -> U112^1(X1, X2, X3)
   U112^1(X1, X2, active(X3)) -> U112^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(281) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(282)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U112^1(X1, mark(X2), X3) -> U112^1(X1, X2, X3)
   U112^1(mark(X1), X2, X3) -> U112^1(X1, X2, X3)
   U112^1(X1, X2, mark(X3)) -> U112^1(X1, X2, X3)
   U112^1(active(X1), X2, X3) -> U112^1(X1, X2, X3)
   U112^1(X1, active(X2), X3) -> U112^1(X1, X2, X3)
   U112^1(X1, X2, active(X3)) -> U112^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(283) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U112^1(X1, mark(X2), X3) -> U112^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U112^1(mark(X1), X2, X3) -> U112^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U112^1(X1, X2, mark(X3)) -> U112^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U112^1(active(X1), X2, X3) -> U112^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U112^1(X1, active(X2), X3) -> U112^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U112^1(X1, X2, active(X3)) -> U112^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(284)
YES

----------------------------------------

(285)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U111^1(X1, mark(X2), X3) -> U111^1(X1, X2, X3)
   U111^1(mark(X1), X2, X3) -> U111^1(X1, X2, X3)
   U111^1(X1, X2, mark(X3)) -> U111^1(X1, X2, X3)
   U111^1(active(X1), X2, X3) -> U111^1(X1, X2, X3)
   U111^1(X1, active(X2), X3) -> U111^1(X1, X2, X3)
   U111^1(X1, X2, active(X3)) -> U111^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(286) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(287)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U111^1(X1, mark(X2), X3) -> U111^1(X1, X2, X3)
   U111^1(mark(X1), X2, X3) -> U111^1(X1, X2, X3)
   U111^1(X1, X2, mark(X3)) -> U111^1(X1, X2, X3)
   U111^1(active(X1), X2, X3) -> U111^1(X1, X2, X3)
   U111^1(X1, active(X2), X3) -> U111^1(X1, X2, X3)
   U111^1(X1, X2, active(X3)) -> U111^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(288) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(289)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U111^1(X1, mark(X2), X3) -> U111^1(X1, X2, X3)
   U111^1(mark(X1), X2, X3) -> U111^1(X1, X2, X3)
   U111^1(X1, X2, mark(X3)) -> U111^1(X1, X2, X3)
   U111^1(active(X1), X2, X3) -> U111^1(X1, X2, X3)
   U111^1(X1, active(X2), X3) -> U111^1(X1, X2, X3)
   U111^1(X1, X2, active(X3)) -> U111^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(290) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U111^1(X1, mark(X2), X3) -> U111^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U111^1(mark(X1), X2, X3) -> U111^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U111^1(X1, X2, mark(X3)) -> U111^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U111^1(active(X1), X2, X3) -> U111^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U111^1(X1, active(X2), X3) -> U111^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U111^1(X1, X2, active(X3)) -> U111^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(291)
YES

----------------------------------------

(292)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U12^1(X1, mark(X2)) -> U12^1(X1, X2)
   U12^1(mark(X1), X2) -> U12^1(X1, X2)
   U12^1(active(X1), X2) -> U12^1(X1, X2)
   U12^1(X1, active(X2)) -> U12^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(293) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(294)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U12^1(X1, mark(X2)) -> U12^1(X1, X2)
   U12^1(mark(X1), X2) -> U12^1(X1, X2)
   U12^1(active(X1), X2) -> U12^1(X1, X2)
   U12^1(X1, active(X2)) -> U12^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(295) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(296)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U12^1(X1, mark(X2)) -> U12^1(X1, X2)
   U12^1(mark(X1), X2) -> U12^1(X1, X2)
   U12^1(active(X1), X2) -> U12^1(X1, X2)
   U12^1(X1, active(X2)) -> U12^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(297) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U12^1(X1, mark(X2)) -> U12^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U12^1(mark(X1), X2) -> U12^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U12^1(active(X1), X2) -> U12^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U12^1(X1, active(X2)) -> U12^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(298)
YES

----------------------------------------

(299)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
   U11^1(mark(X1), X2) -> U11^1(X1, X2)
   U11^1(active(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, active(X2)) -> U11^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(300) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(301)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
   U11^1(mark(X1), X2) -> U11^1(X1, X2)
   U11^1(active(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, active(X2)) -> U11^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(302) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(303)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U11^1(X1, mark(X2)) -> U11^1(X1, X2)
   U11^1(mark(X1), X2) -> U11^1(X1, X2)
   U11^1(active(X1), X2) -> U11^1(X1, X2)
   U11^1(X1, active(X2)) -> U11^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(304) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U11^1(X1, mark(X2)) -> U11^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U11^1(mark(X1), X2) -> U11^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U11^1(active(X1), X2) -> U11^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U11^1(X1, active(X2)) -> U11^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(305)
YES

----------------------------------------

(306)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(active(X)) -> ISNATILIST(X)
   ISNATILIST(mark(X)) -> ISNATILIST(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(307) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(308)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(active(X)) -> ISNATILIST(X)
   ISNATILIST(mark(X)) -> ISNATILIST(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(309) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(310)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILIST(active(X)) -> ISNATILIST(X)
   ISNATILIST(mark(X)) -> ISNATILIST(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(311) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*ISNATILIST(active(X)) -> ISNATILIST(X)
The graph contains the following edges 1 > 1


*ISNATILIST(mark(X)) -> ISNATILIST(X)
The graph contains the following edges 1 > 1


----------------------------------------

(312)
YES

----------------------------------------

(313)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U106^1(active(X)) -> U106^1(X)
   U106^1(mark(X)) -> U106^1(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(314) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(315)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U106^1(active(X)) -> U106^1(X)
   U106^1(mark(X)) -> U106^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(316) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(317)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U106^1(active(X)) -> U106^1(X)
   U106^1(mark(X)) -> U106^1(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(318) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U106^1(active(X)) -> U106^1(X)
The graph contains the following edges 1 > 1


*U106^1(mark(X)) -> U106^1(X)
The graph contains the following edges 1 > 1


----------------------------------------

(319)
YES

----------------------------------------

(320)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(active(X)) -> ISNAT(X)
   ISNAT(mark(X)) -> ISNAT(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(321) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(322)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(active(X)) -> ISNAT(X)
   ISNAT(mark(X)) -> ISNAT(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(323) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(324)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNAT(active(X)) -> ISNAT(X)
   ISNAT(mark(X)) -> ISNAT(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(325) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*ISNAT(active(X)) -> ISNAT(X)
The graph contains the following edges 1 > 1


*ISNAT(mark(X)) -> ISNAT(X)
The graph contains the following edges 1 > 1


----------------------------------------

(326)
YES

----------------------------------------

(327)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U105^1(X1, mark(X2)) -> U105^1(X1, X2)
   U105^1(mark(X1), X2) -> U105^1(X1, X2)
   U105^1(active(X1), X2) -> U105^1(X1, X2)
   U105^1(X1, active(X2)) -> U105^1(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(328) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(329)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U105^1(X1, mark(X2)) -> U105^1(X1, X2)
   U105^1(mark(X1), X2) -> U105^1(X1, X2)
   U105^1(active(X1), X2) -> U105^1(X1, X2)
   U105^1(X1, active(X2)) -> U105^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(330) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(331)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U105^1(X1, mark(X2)) -> U105^1(X1, X2)
   U105^1(mark(X1), X2) -> U105^1(X1, X2)
   U105^1(active(X1), X2) -> U105^1(X1, X2)
   U105^1(X1, active(X2)) -> U105^1(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(332) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U105^1(X1, mark(X2)) -> U105^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*U105^1(mark(X1), X2) -> U105^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U105^1(active(X1), X2) -> U105^1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*U105^1(X1, active(X2)) -> U105^1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(333)
YES

----------------------------------------

(334)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3)
   U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(335) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(336)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3)
   U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(337) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(338)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3)
   U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3)
   U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3)
   U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(339) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U104^1(X1, mark(X2), X3) -> U104^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U104^1(mark(X1), X2, X3) -> U104^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U104^1(X1, X2, mark(X3)) -> U104^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U104^1(active(X1), X2, X3) -> U104^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U104^1(X1, active(X2), X3) -> U104^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U104^1(X1, X2, active(X3)) -> U104^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(340)
YES

----------------------------------------

(341)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(active(X)) -> ISNATILISTKIND(X)
   ISNATILISTKIND(mark(X)) -> ISNATILISTKIND(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(342) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(343)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(active(X)) -> ISNATILISTKIND(X)
   ISNATILISTKIND(mark(X)) -> ISNATILISTKIND(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(344) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(345)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATILISTKIND(active(X)) -> ISNATILISTKIND(X)
   ISNATILISTKIND(mark(X)) -> ISNATILISTKIND(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(346) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*ISNATILISTKIND(active(X)) -> ISNATILISTKIND(X)
The graph contains the following edges 1 > 1


*ISNATILISTKIND(mark(X)) -> ISNATILISTKIND(X)
The graph contains the following edges 1 > 1


----------------------------------------

(347)
YES

----------------------------------------

(348)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3)
   U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(349) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(350)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3)
   U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(351) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(352)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3)
   U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3)
   U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3)
   U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(353) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U103^1(X1, mark(X2), X3) -> U103^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U103^1(mark(X1), X2, X3) -> U103^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U103^1(X1, X2, mark(X3)) -> U103^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U103^1(active(X1), X2, X3) -> U103^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U103^1(X1, active(X2), X3) -> U103^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U103^1(X1, X2, active(X3)) -> U103^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(354)
YES

----------------------------------------

(355)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATKIND(active(X)) -> ISNATKIND(X)
   ISNATKIND(mark(X)) -> ISNATKIND(X)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(356) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(357)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATKIND(active(X)) -> ISNATKIND(X)
   ISNATKIND(mark(X)) -> ISNATKIND(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(358) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(359)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ISNATKIND(active(X)) -> ISNATKIND(X)
   ISNATKIND(mark(X)) -> ISNATKIND(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(360) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*ISNATKIND(active(X)) -> ISNATKIND(X)
The graph contains the following edges 1 > 1


*ISNATKIND(mark(X)) -> ISNATKIND(X)
The graph contains the following edges 1 > 1


----------------------------------------

(361)
YES

----------------------------------------

(362)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3)
   U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(363) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(364)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3)
   U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(365) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(366)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3)
   U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3)
   U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3)
   U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(367) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U102^1(X1, mark(X2), X3) -> U102^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U102^1(mark(X1), X2, X3) -> U102^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U102^1(X1, X2, mark(X3)) -> U102^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U102^1(active(X1), X2, X3) -> U102^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U102^1(X1, active(X2), X3) -> U102^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U102^1(X1, X2, active(X3)) -> U102^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(368)
YES

----------------------------------------

(369)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3)
   U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(370) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(371)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3)
   U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(372) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(373)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3)
   U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3)
   U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3)
   U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(374) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U101^1(X1, mark(X2), X3) -> U101^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U101^1(mark(X1), X2, X3) -> U101^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U101^1(X1, X2, mark(X3)) -> U101^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*U101^1(active(X1), X2, X3) -> U101^1(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*U101^1(X1, active(X2), X3) -> U101^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*U101^1(X1, X2, active(X3)) -> U101^1(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


----------------------------------------

(375)
YES

----------------------------------------

(376)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   CONS(X1, mark(X2)) -> CONS(X1, X2)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(active(X1), X2) -> CONS(X1, X2)
   CONS(X1, active(X2)) -> CONS(X1, X2)

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(377) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(378)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   CONS(X1, mark(X2)) -> CONS(X1, X2)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(active(X1), X2) -> CONS(X1, X2)
   CONS(X1, active(X2)) -> CONS(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(379) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(380)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   CONS(X1, mark(X2)) -> CONS(X1, X2)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(active(X1), X2) -> CONS(X1, X2)
   CONS(X1, active(X2)) -> CONS(X1, X2)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(381) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*CONS(X1, mark(X2)) -> CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*CONS(mark(X1), X2) -> CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*CONS(active(X1), X2) -> CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*CONS(X1, active(X2)) -> CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(382)
YES

----------------------------------------

(383)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> ACTIVE(U106(mark(X)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> ACTIVE(U13(mark(X)))
   ACTIVE(U121(tt, IL)) -> MARK(U122(isNatIListKind(IL)))
   MARK(U122(X)) -> ACTIVE(U122(mark(X)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> ACTIVE(U33(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> ACTIVE(U46(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> ACTIVE(U96(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(take(0, IL)) -> MARK(U121(isNatIList(IL), IL))
   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(U131(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> ACTIVE(U71(mark(X)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U121(X1, X2)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(U132(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U133(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U134(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U135(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U136(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> ACTIVE(U81(mark(X)))
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   mark(0) -> active(0)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   mark(tt) -> active(tt)
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(U106(X)) -> active(U106(mark(X)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   mark(s(X)) -> active(s(mark(X)))
   mark(length(X)) -> active(length(mark(X)))
   mark(U13(X)) -> active(U13(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   mark(U122(X)) -> active(U122(mark(X)))
   mark(nil) -> active(nil)
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   mark(U23(X)) -> active(U23(mark(X)))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   mark(U33(X)) -> active(U33(mark(X)))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   mark(U46(X)) -> active(U46(mark(X)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   mark(U52(X)) -> active(U52(mark(X)))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   mark(U62(X)) -> active(U62(mark(X)))
   mark(U71(X)) -> active(U71(mark(X)))
   mark(U81(X)) -> active(U81(mark(X)))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   mark(U96(X)) -> active(U96(mark(X)))
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatKind(active(X)) -> isNatKind(X)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)
   U106(mark(X)) -> U106(X)
   U106(active(X)) -> U106(X)
   isNatIList(mark(X)) -> isNatIList(X)
   isNatIList(active(X)) -> isNatIList(X)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   length(mark(X)) -> length(X)
   length(active(X)) -> length(X)
   U13(mark(X)) -> U13(X)
   U13(active(X)) -> U13(X)
   isNatList(mark(X)) -> isNatList(X)
   isNatList(active(X)) -> isNatList(X)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U122(mark(X)) -> U122(X)
   U122(active(X)) -> U122(X)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   take(mark(X1), X2) -> take(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(mark(X)) -> U23(X)
   U23(active(X)) -> U23(X)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(mark(X)) -> U33(X)
   U33(active(X)) -> U33(X)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(mark(X)) -> U46(X)
   U46(active(X)) -> U46(X)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U52(mark(X)) -> U52(X)
   U52(active(X)) -> U52(X)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U62(mark(X)) -> U62(X)
   U62(active(X)) -> U62(X)
   U71(mark(X)) -> U71(X)
   U71(active(X)) -> U71(X)
   U81(mark(X)) -> U81(X)
   U81(active(X)) -> U81(X)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(mark(X)) -> U96(X)
   U96(active(X)) -> U96(X)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(384) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(385)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> ACTIVE(U106(mark(X)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> ACTIVE(U13(mark(X)))
   ACTIVE(U121(tt, IL)) -> MARK(U122(isNatIListKind(IL)))
   MARK(U122(X)) -> ACTIVE(U122(mark(X)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> ACTIVE(U33(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> ACTIVE(U46(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> ACTIVE(U96(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(take(0, IL)) -> MARK(U121(isNatIList(IL), IL))
   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(U131(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> ACTIVE(U71(mark(X)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U121(X1, X2)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(U132(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U133(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U134(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U135(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U136(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> ACTIVE(U81(mark(X)))
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(386) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U106(X)) -> ACTIVE(U106(mark(X)))
   MARK(U13(X)) -> ACTIVE(U13(mark(X)))
   MARK(U122(X)) -> ACTIVE(U122(mark(X)))
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   MARK(U33(X)) -> ACTIVE(U33(mark(X)))
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   MARK(U96(X)) -> ACTIVE(U96(mark(X)))
   MARK(U71(X)) -> ACTIVE(U71(mark(X)))
   MARK(U81(X)) -> ACTIVE(U81(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U106_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 0
POL( U13_1(x_1) ) = max{0, -2}
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = 0
POL( U81_1(x_1) ) = 0
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( U96_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = 2
POL( take_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = 2
POL( zeros ) = 2
POL( active_1(x_1) ) = 2
POL( 0 ) = 0
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


----------------------------------------

(387)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U121(tt, IL)) -> MARK(U122(isNatIListKind(IL)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> ACTIVE(U46(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(take(0, IL)) -> MARK(U121(isNatIList(IL), IL))
   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(U131(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U121(X1, X2)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(U132(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U133(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U134(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U135(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U136(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(388) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   MARK(U46(X)) -> ACTIVE(U46(mark(X)))
   MARK(U62(X)) -> ACTIVE(U62(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U121_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = 2
POL( take_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, 2x_1 - 1}
POL( 0 ) = 2
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 2
POL( U122_1(x_1) ) = max{0, x_1 - 1}
POL( U23_1(x_1) ) = 2x_1 + 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U52_1(x_1) ) = 1
POL( U96_1(x_1) ) = 2
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(389)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U121(tt, IL)) -> MARK(U122(isNatIListKind(IL)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(take(0, IL)) -> MARK(U121(isNatIList(IL), IL))
   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(U131(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U121(X1, X2)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(U132(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U133(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U134(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U135(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U136(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(390) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(s(X)) -> ACTIVE(s(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U121_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = 0
POL( take_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = 0
POL( zeros ) = 2
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( 0 ) = 0
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 2
POL( U122_1(x_1) ) = 2
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = 1
POL( U46_1(x_1) ) = 1
POL( U52_1(x_1) ) = 1
POL( U62_1(x_1) ) = 2
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = 2x_1 + 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(391)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U121(tt, IL)) -> MARK(U122(isNatIListKind(IL)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(take(0, IL)) -> MARK(U121(isNatIList(IL), IL))
   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(U131(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U121(X1, X2)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(U132(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U133(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U134(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U135(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U136(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(392) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(take(0, IL)) -> MARK(U121(isNatIList(IL), IL))
   MARK(U131(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U132(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U133(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U134(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U135(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U136(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = x_1
   POL(U102(x_1, x_2, x_3)) = x_1
   POL(U103(x_1, x_2, x_3)) = x_1
   POL(U104(x_1, x_2, x_3)) = x_1
   POL(U105(x_1, x_2)) = x_1
   POL(U106(x_1)) = x_1
   POL(U11(x_1, x_2)) = x_1
   POL(U111(x_1, x_2, x_3)) = x_1 + x_2
   POL(U112(x_1, x_2, x_3)) = x_1 + x_2
   POL(U113(x_1, x_2, x_3)) = x_1 + x_2
   POL(U114(x_1, x_2)) = x_1 + x_2
   POL(U12(x_1, x_2)) = x_1
   POL(U121(x_1, x_2)) = x_1
   POL(U122(x_1)) = x_1
   POL(U13(x_1)) = x_1
   POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
   POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
   POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
   POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
   POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1, x_2)) = x_1
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1
   POL(U32(x_1, x_2)) = x_1
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = x_1
   POL(U42(x_1, x_2, x_3)) = x_1
   POL(U43(x_1, x_2, x_3)) = x_1
   POL(U44(x_1, x_2, x_3)) = x_1
   POL(U45(x_1, x_2)) = x_1
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = x_1
   POL(U92(x_1, x_2, x_3)) = x_1
   POL(U93(x_1, x_2, x_3)) = x_1
   POL(U94(x_1, x_2, x_3)) = x_1
   POL(U95(x_1, x_2)) = x_1
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 1 + x_1 + x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


----------------------------------------

(393)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U121(tt, IL)) -> MARK(U122(isNatIListKind(IL)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U121(X1, X2)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(394) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U121(tt, IL)) -> MARK(U122(isNatIListKind(IL)))
   MARK(U121(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = x_1
   POL(U102(x_1, x_2, x_3)) = x_1
   POL(U103(x_1, x_2, x_3)) = x_1
   POL(U104(x_1, x_2, x_3)) = x_1
   POL(U105(x_1, x_2)) = x_1
   POL(U106(x_1)) = x_1
   POL(U11(x_1, x_2)) = x_1
   POL(U111(x_1, x_2, x_3)) = x_1 + x_2
   POL(U112(x_1, x_2, x_3)) = x_1 + x_2
   POL(U113(x_1, x_2, x_3)) = x_1 + x_2
   POL(U114(x_1, x_2)) = x_1 + x_2
   POL(U12(x_1, x_2)) = x_1
   POL(U121(x_1, x_2)) = 1 + x_1
   POL(U122(x_1)) = x_1
   POL(U13(x_1)) = x_1
   POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_4
   POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_4
   POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_4
   POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_4
   POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_4
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1, x_2)) = x_1
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1
   POL(U32(x_1, x_2)) = x_1
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = x_1
   POL(U42(x_1, x_2, x_3)) = x_1
   POL(U43(x_1, x_2, x_3)) = x_1
   POL(U44(x_1, x_2, x_3)) = x_1
   POL(U45(x_1, x_2)) = x_1
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = x_1
   POL(U92(x_1, x_2, x_3)) = x_1
   POL(U93(x_1, x_2, x_3)) = x_1
   POL(U94(x_1, x_2, x_3)) = x_1
   POL(U95(x_1, x_2)) = x_1
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 1 + x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


----------------------------------------

(395)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(396) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U121_2(x_1, x_2) ) = 0
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( take_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = 2
POL( zeros ) = 2
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( 0 ) = 0
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 1
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 2
POL( U122_1(x_1) ) = 1
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( U52_1(x_1) ) = x_1 + 1
POL( U62_1(x_1) ) = x_1
POL( U96_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(397)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(length(X)) -> MARK(X)
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(398) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(length(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = x_1
   POL(U102(x_1, x_2, x_3)) = x_1
   POL(U103(x_1, x_2, x_3)) = x_1
   POL(U104(x_1, x_2, x_3)) = x_1
   POL(U105(x_1, x_2)) = x_1
   POL(U106(x_1)) = x_1
   POL(U11(x_1, x_2)) = x_1
   POL(U111(x_1, x_2, x_3)) = 1 + x_1 + x_2
   POL(U112(x_1, x_2, x_3)) = 1 + x_1 + x_2
   POL(U113(x_1, x_2, x_3)) = 1 + x_1 + x_2
   POL(U114(x_1, x_2)) = 1 + x_1 + x_2
   POL(U12(x_1, x_2)) = x_1
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = x_1
   POL(U13(x_1)) = x_1
   POL(U131(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U133(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U134(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U135(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U136(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1, x_2)) = x_1
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1
   POL(U32(x_1, x_2)) = x_1
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = x_1
   POL(U42(x_1, x_2, x_3)) = x_1
   POL(U43(x_1, x_2, x_3)) = x_1
   POL(U44(x_1, x_2, x_3)) = x_1
   POL(U45(x_1, x_2)) = x_1
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = x_1
   POL(U92(x_1, x_2, x_3)) = x_1
   POL(U93(x_1, x_2, x_3)) = x_1
   POL(U94(x_1, x_2, x_3)) = x_1
   POL(U95(x_1, x_2)) = x_1
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 1 + x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)


----------------------------------------

(399)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(400) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U102(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(U103(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U104(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U105(x_1, x_2)) = x_1 + x_2
   POL(U106(x_1)) = x_1
   POL(U11(x_1, x_2)) = x_1 + x_2
   POL(U111(x_1, x_2, x_3)) = x_2
   POL(U112(x_1, x_2, x_3)) = x_2
   POL(U113(x_1, x_2, x_3)) = x_2
   POL(U114(x_1, x_2)) = x_2
   POL(U12(x_1, x_2)) = x_1 + x_2
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = x_1
   POL(U13(x_1)) = x_1
   POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U21(x_1, x_2)) = x_1 + x_2
   POL(U22(x_1, x_2)) = x_1 + x_2
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1 + x_2
   POL(U32(x_1, x_2)) = x_1 + x_2
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U42(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U43(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U44(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U45(x_1, x_2)) = x_1 + x_2
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U92(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U93(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U94(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U95(x_1, x_2)) = x_1 + x_2
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = x_1
   POL(isNatIList(x_1)) = x_1
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = x_1
   POL(length(x_1)) = x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 1 + x_1 + x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)


----------------------------------------

(401)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(402) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( take_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = 2
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( 0 ) = 2
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 1
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = x_1 + 2
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 2
POL( U23_1(x_1) ) = max{0, -2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = 2
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U96_1(x_1) ) = 2x_1 + 2
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = x_1 + 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(403)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U101(tt, V1, V2)) -> MARK(U102(isNatKind(V1), V1, V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U122(X)) -> MARK(X)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(404) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(405)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U122(X)) -> MARK(X)
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(406) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U101(X1, X2, X3)) -> ACTIVE(U101(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U101_3(x_1, ..., x_3) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( take_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( 0 ) = 2
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = x_1 + 2
POL( U122_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = 2x_1 + 1
POL( U96_1(x_1) ) = 2
POL( U71_1(x_1) ) = 2x_1
POL( U81_1(x_1) ) = 2x_1
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(407)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U122(X)) -> MARK(X)
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(408) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(409)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U122(X)) -> MARK(X)
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(410) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = 0
   POL(U102(x_1, x_2, x_3)) = 0
   POL(U103(x_1, x_2, x_3)) = x_1
   POL(U104(x_1, x_2, x_3)) = x_1
   POL(U105(x_1, x_2)) = x_1
   POL(U106(x_1)) = x_1
   POL(U11(x_1, x_2)) = x_1
   POL(U111(x_1, x_2, x_3)) = 0
   POL(U112(x_1, x_2, x_3)) = 0
   POL(U113(x_1, x_2, x_3)) = 0
   POL(U114(x_1, x_2)) = 0
   POL(U12(x_1, x_2)) = x_1
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = x_1
   POL(U13(x_1)) = x_1
   POL(U131(x_1, x_2, x_3, x_4)) = x_4
   POL(U132(x_1, x_2, x_3, x_4)) = x_4
   POL(U133(x_1, x_2, x_3, x_4)) = x_4
   POL(U134(x_1, x_2, x_3, x_4)) = x_4
   POL(U135(x_1, x_2, x_3, x_4)) = x_4
   POL(U136(x_1, x_2, x_3, x_4)) = x_4
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1, x_2)) = x_1
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1
   POL(U32(x_1, x_2)) = x_1
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = x_1
   POL(U42(x_1, x_2, x_3)) = x_1
   POL(U43(x_1, x_2, x_3)) = x_1
   POL(U44(x_1, x_2, x_3)) = x_1
   POL(U45(x_1, x_2)) = x_1
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = x_1
   POL(U92(x_1, x_2, x_3)) = x_1
   POL(U93(x_1, x_2, x_3)) = x_1
   POL(U94(x_1, x_2, x_3)) = x_1
   POL(U95(x_1, x_2)) = x_1
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 1 + x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)


----------------------------------------

(411)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U122(X)) -> MARK(X)
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(412) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( take_2(x_1, x_2) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = x_2 + 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = x_1 + 2
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U122_1(x_1) ) = max{0, x_1 - 2}
POL( U23_1(x_1) ) = x_1 + 2
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U46_1(x_1) ) = 2
POL( U52_1(x_1) ) = 0
POL( U62_1(x_1) ) = x_1 + 2
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 1
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(413)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U122(X)) -> MARK(X)
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(414) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = 0
   POL(U102(x_1, x_2, x_3)) = 0
   POL(U103(x_1, x_2, x_3)) = x_1
   POL(U104(x_1, x_2, x_3)) = x_1
   POL(U105(x_1, x_2)) = x_1
   POL(U106(x_1)) = x_1
   POL(U11(x_1, x_2)) = x_1
   POL(U111(x_1, x_2, x_3)) = 0
   POL(U112(x_1, x_2, x_3)) = 0
   POL(U113(x_1, x_2, x_3)) = 0
   POL(U114(x_1, x_2)) = 0
   POL(U12(x_1, x_2)) = x_1
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = x_1
   POL(U13(x_1)) = x_1
   POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = x_4
   POL(U133(x_1, x_2, x_3, x_4)) = x_4
   POL(U134(x_1, x_2, x_3, x_4)) = x_4
   POL(U135(x_1, x_2, x_3, x_4)) = x_4
   POL(U136(x_1, x_2, x_3, x_4)) = x_4
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1, x_2)) = x_1
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1
   POL(U32(x_1, x_2)) = x_1
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = x_1
   POL(U42(x_1, x_2, x_3)) = x_1
   POL(U43(x_1, x_2, x_3)) = x_1
   POL(U44(x_1, x_2, x_3)) = x_1
   POL(U45(x_1, x_2)) = x_1
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = x_1
   POL(U92(x_1, x_2, x_3)) = x_1
   POL(U93(x_1, x_2, x_3)) = x_1
   POL(U94(x_1, x_2, x_3)) = x_1
   POL(U95(x_1, x_2)) = x_1
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 1 + x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(415)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U122(X)) -> MARK(X)
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(416) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, -2}
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U122_1(x_1) ) = max{0, -2}
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = 2x_1 + 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 0
POL( U52_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = 2
POL( U96_1(x_1) ) = x_1
POL( U71_1(x_1) ) = max{0, x_1 - 1}
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(417)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U122(X)) -> MARK(X)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(418) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U122(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = 0
   POL(U102(x_1, x_2, x_3)) = 0
   POL(U103(x_1, x_2, x_3)) = x_1
   POL(U104(x_1, x_2, x_3)) = x_1
   POL(U105(x_1, x_2)) = x_1
   POL(U106(x_1)) = x_1
   POL(U11(x_1, x_2)) = x_1
   POL(U111(x_1, x_2, x_3)) = 0
   POL(U112(x_1, x_2, x_3)) = 0
   POL(U113(x_1, x_2, x_3)) = 0
   POL(U114(x_1, x_2)) = 0
   POL(U12(x_1, x_2)) = x_1
   POL(U121(x_1, x_2)) = 1
   POL(U122(x_1)) = 1 + x_1
   POL(U13(x_1)) = x_1
   POL(U131(x_1, x_2, x_3, x_4)) = x_4
   POL(U132(x_1, x_2, x_3, x_4)) = x_4
   POL(U133(x_1, x_2, x_3, x_4)) = x_4
   POL(U134(x_1, x_2, x_3, x_4)) = x_4
   POL(U135(x_1, x_2, x_3, x_4)) = x_4
   POL(U136(x_1, x_2, x_3, x_4)) = x_4
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1, x_2)) = x_1
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1
   POL(U32(x_1, x_2)) = x_1
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = x_1
   POL(U42(x_1, x_2, x_3)) = x_1
   POL(U43(x_1, x_2, x_3)) = x_1
   POL(U44(x_1, x_2, x_3)) = x_1
   POL(U45(x_1, x_2)) = x_1
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = x_1
   POL(U92(x_1, x_2, x_3)) = x_1
   POL(U93(x_1, x_2, x_3)) = x_1
   POL(U94(x_1, x_2, x_3)) = x_1
   POL(U95(x_1, x_2)) = x_1
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 1 + x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(419)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(420) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = 0
   POL(U102(x_1, x_2, x_3)) = 0
   POL(U103(x_1, x_2, x_3)) = x_1
   POL(U104(x_1, x_2, x_3)) = x_1
   POL(U105(x_1, x_2)) = x_1
   POL(U106(x_1)) = x_1
   POL(U11(x_1, x_2)) = x_1
   POL(U111(x_1, x_2, x_3)) = 0
   POL(U112(x_1, x_2, x_3)) = 0
   POL(U113(x_1, x_2, x_3)) = 0
   POL(U114(x_1, x_2)) = 0
   POL(U12(x_1, x_2)) = x_1
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = 0
   POL(U13(x_1)) = x_1
   POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_4
   POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_4
   POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_4
   POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_4
   POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_4
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1, x_2)) = x_1
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_1
   POL(U32(x_1, x_2)) = x_1
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = x_1
   POL(U42(x_1, x_2, x_3)) = x_1
   POL(U43(x_1, x_2, x_3)) = x_1
   POL(U44(x_1, x_2, x_3)) = x_1
   POL(U45(x_1, x_2)) = x_1
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = x_1
   POL(U92(x_1, x_2, x_3)) = x_1
   POL(U93(x_1, x_2, x_3)) = x_1
   POL(U94(x_1, x_2, x_3)) = x_1
   POL(U95(x_1, x_2)) = x_1
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 1 + x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(421)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(422) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2x_1 + 2
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U122_1(x_1) ) = 1
POL( U131_4(x_1, ..., x_4) ) = x_1 + x_3 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = 0
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( U52_1(x_1) ) = 0
POL( U62_1(x_1) ) = 0
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = max{0, 2x_1 - 1}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(423)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(424) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(425)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(426) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 0
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = 0
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = 2
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + x_4 + 2
POL( take_2(x_1, x_2) ) = 2x_1 + 2
POL( U23_1(x_1) ) = max{0, -2}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = 0
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( U81_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(427)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(428) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(429)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(430) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 2
POL( U104_3(x_1, ..., x_3) ) = 2x_1
POL( U105_2(x_1, x_2) ) = 2x_1
POL( U11_2(x_1, x_2) ) = 2x_1
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = 2x_1
POL( U132_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, -2}
POL( U134_4(x_1, ..., x_4) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2x_1
POL( U22_2(x_1, x_2) ) = 2x_1
POL( U31_2(x_1, x_2) ) = 2x_1
POL( U32_2(x_1, x_2) ) = 2x_1
POL( U41_3(x_1, ..., x_3) ) = 2x_1
POL( U42_3(x_1, ..., x_3) ) = 2x_1
POL( U43_3(x_1, ..., x_3) ) = 2x_1
POL( U44_3(x_1, ..., x_3) ) = x_1
POL( U45_2(x_1, x_2) ) = 2x_1
POL( U51_2(x_1, x_2) ) = 2x_1
POL( U61_2(x_1, x_2) ) = 2x_1
POL( U91_3(x_1, ..., x_3) ) = 2x_1
POL( U92_3(x_1, ..., x_3) ) = 2x_1
POL( U93_3(x_1, ..., x_3) ) = x_1
POL( U94_3(x_1, ..., x_3) ) = x_1
POL( U95_2(x_1, x_2) ) = 2x_1
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2x_1
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3 + 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( isNatIListKind_1(x_1) ) = 0
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1
POL( U13_1(x_1) ) = x_1
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + x_3 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = x_1
POL( U33_1(x_1) ) = x_1
POL( U46_1(x_1) ) = 2x_1
POL( U52_1(x_1) ) = x_1
POL( U62_1(x_1) ) = x_1
POL( U96_1(x_1) ) = x_1
POL( U71_1(x_1) ) = x_1
POL( U81_1(x_1) ) = x_1
POL( nil ) = 0
POL( MARK_1(x_1) ) = x_1 + 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(431)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(432) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, -2}
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 0
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 - 1}
POL( tt ) = 1
POL( U102_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U122_1(x_1) ) = 0
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U23_1(x_1) ) = 1
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = 2
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 1
POL( U81_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, -2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(433)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(434) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(zeros) -> ACTIVE(zeros)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( U105_2(x_1, x_2) ) = 2x_1 + 1
POL( U11_2(x_1, x_2) ) = 2x_1 + 1
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U113_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = 2x_1 + 1
POL( U133_4(x_1, ..., x_4) ) = max{0, -2}
POL( U134_4(x_1, ..., x_4) ) = x_1 + 1
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = 2x_1 + 1
POL( U32_2(x_1, x_2) ) = 2x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = x_1 + 1
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U45_2(x_1, x_2) ) = x_1 + 1
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U95_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( zeros ) = 2
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = 2x_1 + 1
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( U13_1(x_1) ) = 2x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = max{0, -2}
POL( U131_4(x_1, ..., x_4) ) = 2x_1 + x_3 + 2
POL( U132_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 + x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = x_2 + 2
POL( take_2(x_1, x_2) ) = 2x_2 + 2
POL( U23_1(x_1) ) = x_1 + 1
POL( U33_1(x_1) ) = x_1 + 1
POL( U46_1(x_1) ) = 2x_1 + 1
POL( U52_1(x_1) ) = x_1 + 1
POL( U62_1(x_1) ) = 2x_1 + 1
POL( U96_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(435)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(zeros) -> MARK(cons(0, zeros))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(436) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(437)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(438) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   ACTIVE(U132(tt, IL, M, N)) -> MARK(U133(isNat(M), IL, M, N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( U105_2(x_1, x_2) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 0
POL( U51_2(x_1, x_2) ) = 0
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U13_1(x_1) ) = max{0, x_1 - 2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, x_3 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 + x_4 - 2}
POL( take_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 0
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U96_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(439)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(440) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U105_2(x_1, x_2) ) = 2x_1 + 2
POL( U11_2(x_1, x_2) ) = 2x_1 + 1
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 0
POL( U12_2(x_1, x_2) ) = 2x_1 + 1
POL( U133_4(x_1, ..., x_4) ) = x_2 + 2x_3 + x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = 1
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = 2x_1 + 1
POL( U32_2(x_1, x_2) ) = 2x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = x_1 + 1
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U45_2(x_1, x_2) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = x_1 + 1
POL( U95_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( active_1(x_1) ) = 1
POL( cons_2(x_1, x_2) ) = x_1 + 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = x_2 + x_3 + 1
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( U13_1(x_1) ) = 2x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U122_1(x_1) ) = max{0, -2}
POL( U131_4(x_1, ..., x_4) ) = max{0, x_2 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_2 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_2 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, x_2 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U33_1(x_1) ) = 2x_1 + 1
POL( U46_1(x_1) ) = 2x_1 + 1
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U62_1(x_1) ) = 2x_1 + 1
POL( U96_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U81_1(x_1) ) = x_1 + 1
POL( nil ) = 2
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(441)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U106(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(442) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = 1
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U134_4(x_1, ..., x_4) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = 0
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 0
POL( U42_3(x_1, ..., x_3) ) = 1
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 0
POL( U61_2(x_1, x_2) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 1
POL( U93_3(x_1, ..., x_3) ) = 0
POL( U94_3(x_1, ..., x_3) ) = 1
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 1
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( 0 ) = 1
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = x_1
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 1
POL( s_1(x_1) ) = x_1
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 + x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_1 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( take_2(x_1, x_2) ) = 2x_1 + x_2 + 1
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U46_1(x_1) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = 2
POL( U96_1(x_1) ) = 2x_1 + 2
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, -2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(443)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U106(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(444) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U134_4(x_1, ..., x_4) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 1
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U105_2(x_1, x_2) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = x_1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = x_1
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + x_4 + 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = x_1 + 2
POL( take_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U46_1(x_1) ) = max{0, -2}
POL( U52_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = 1
POL( U96_1(x_1) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = max{0, x_1 - 1}
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(445)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U106(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(446) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U106(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U11_2(x_1, x_2) ) = x_1 + 1
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U114_2(x_1, x_2) ) = 0
POL( U12_2(x_1, x_2) ) = 2x_1 + 1
POL( U21_2(x_1, x_2) ) = x_1 + 1
POL( U22_2(x_1, x_2) ) = x_1 + 1
POL( U31_2(x_1, x_2) ) = 2x_1 + 1
POL( U32_2(x_1, x_2) ) = 2x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = x_1 + 1
POL( U45_2(x_1, x_2) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U95_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = 2
POL( zeros ) = 1
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = x_1 + x_2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U105_2(x_1, x_2) ) = 1
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 2x_1 + 2
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1 + 1
POL( U13_1(x_1) ) = 2x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = x_1 + x_2
POL( U122_1(x_1) ) = x_1 + 1
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U133_4(x_1, ..., x_4) ) = x_1 + 2
POL( U134_4(x_1, ..., x_4) ) = x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, -2}
POL( U23_1(x_1) ) = x_1 + 1
POL( U33_1(x_1) ) = 2x_1 + 1
POL( U46_1(x_1) ) = 2x_1 + 1
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U62_1(x_1) ) = x_1 + 1
POL( U96_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(447)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U33(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(448) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = 0
   POL(U102(x_1, x_2, x_3)) = 0
   POL(U103(x_1, x_2, x_3)) = 0
   POL(U104(x_1, x_2, x_3)) = 0
   POL(U105(x_1, x_2)) = 0
   POL(U106(x_1)) = 0
   POL(U11(x_1, x_2)) = x_1
   POL(U111(x_1, x_2, x_3)) = 0
   POL(U112(x_1, x_2, x_3)) = 0
   POL(U113(x_1, x_2, x_3)) = 0
   POL(U114(x_1, x_2)) = 0
   POL(U12(x_1, x_2)) = x_1
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = 0
   POL(U13(x_1)) = x_1
   POL(U131(x_1, x_2, x_3, x_4)) = 0
   POL(U132(x_1, x_2, x_3, x_4)) = 0
   POL(U133(x_1, x_2, x_3, x_4)) = 0
   POL(U134(x_1, x_2, x_3, x_4)) = 0
   POL(U135(x_1, x_2, x_3, x_4)) = 0
   POL(U136(x_1, x_2, x_3, x_4)) = 0
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1, x_2)) = x_1
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = 1 + x_1
   POL(U32(x_1, x_2)) = 1 + x_1
   POL(U33(x_1)) = x_1
   POL(U41(x_1, x_2, x_3)) = 1 + x_1
   POL(U42(x_1, x_2, x_3)) = 1 + x_1
   POL(U43(x_1, x_2, x_3)) = 1 + x_1
   POL(U44(x_1, x_2, x_3)) = 1 + x_1
   POL(U45(x_1, x_2)) = 1 + x_1
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = x_1
   POL(U92(x_1, x_2, x_3)) = x_1
   POL(U93(x_1, x_2, x_3)) = x_1
   POL(U94(x_1, x_2, x_3)) = x_1
   POL(U95(x_1, x_2)) = x_1
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 1
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 0
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(449)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   MARK(U33(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(450) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U33(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = 2x_1 + 1
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 0
POL( U12_2(x_1, x_2) ) = 2x_1 + 1
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = 0
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 0
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U95_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( zeros ) = 1
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1 + 1
POL( U13_1(x_1) ) = 2x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_3 - 2}
POL( U132_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 - 2}
POL( U134_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 - 2}
POL( take_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U23_1(x_1) ) = x_1 + 1
POL( U33_1(x_1) ) = x_1 + 2
POL( U46_1(x_1) ) = 2x_1 + 1
POL( U52_1(x_1) ) = x_1 + 1
POL( U62_1(x_1) ) = x_1 + 1
POL( U96_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = 2x_1 + 1
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(451)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(452) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 1}
POL( U11_2(x_1, x_2) ) = 1
POL( U111_3(x_1, ..., x_3) ) = 1
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U114_2(x_1, x_2) ) = 1
POL( U12_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( U31_2(x_1, x_2) ) = 1
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 1
POL( U42_3(x_1, ..., x_3) ) = 1
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = 1
POL( U51_2(x_1, x_2) ) = 1
POL( U61_2(x_1, x_2) ) = 1
POL( U91_3(x_1, ..., x_3) ) = 1
POL( U92_3(x_1, ..., x_3) ) = 1
POL( U93_3(x_1, ..., x_3) ) = 1
POL( U94_3(x_1, ..., x_3) ) = 1
POL( U95_2(x_1, x_2) ) = 1
POL( length_1(x_1) ) = 1
POL( mark_1(x_1) ) = 2
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 1
POL( U103_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2
POL( isNatIListKind_1(x_1) ) = 1
POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = 2
POL( isNat_1(x_1) ) = 1
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 1
POL( s_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 1
POL( U121_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U122_1(x_1) ) = x_1
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_3 + x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 + x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U46_1(x_1) ) = max{0, -2}
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U62_1(x_1) ) = x_1 + 2
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( U81_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(453)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(454) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(455)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(456) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = max{0, x_1 - 2}
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U122_1(x_1) ) = 1
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_3 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_3 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_3 + x_4 + 1
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 - 2}
POL( take_2(x_1, x_2) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = x_1 + 1
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( U52_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = 0
POL( U96_1(x_1) ) = 0
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(457)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(458) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(459)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(460) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = x_3
   POL(U102(x_1, x_2, x_3)) = 0
   POL(U103(x_1, x_2, x_3)) = 0
   POL(U104(x_1, x_2, x_3)) = 0
   POL(U105(x_1, x_2)) = 0
   POL(U106(x_1)) = 0
   POL(U11(x_1, x_2)) = 1 + x_1 + x_2
   POL(U111(x_1, x_2, x_3)) = 1 + x_2
   POL(U112(x_1, x_2, x_3)) = 1 + x_2
   POL(U113(x_1, x_2, x_3)) = 1 + x_2
   POL(U114(x_1, x_2)) = 1 + x_2
   POL(U12(x_1, x_2)) = 1 + x_1 + x_2
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = 0
   POL(U13(x_1)) = x_1
   POL(U131(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U133(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U134(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U135(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U136(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U21(x_1, x_2)) = x_1 + x_2
   POL(U22(x_1, x_2)) = x_1 + x_2
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = 0
   POL(U32(x_1, x_2)) = 0
   POL(U33(x_1)) = 0
   POL(U41(x_1, x_2, x_3)) = 0
   POL(U42(x_1, x_2, x_3)) = 0
   POL(U43(x_1, x_2, x_3)) = 0
   POL(U44(x_1, x_2, x_3)) = 0
   POL(U45(x_1, x_2)) = 0
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U92(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U93(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U94(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(U95(x_1, x_2)) = x_1 + x_2
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = x_1
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = x_1
   POL(length(x_1)) = 1 + x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(461)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(462) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U13(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U11_2(x_1, x_2) ) = 1
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 1
POL( U51_2(x_1, x_2) ) = x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = x_1 + 1
POL( U95_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = 0
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = x_3 + 2
POL( tt ) = 1
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( U105_2(x_1, x_2) ) = x_2 + 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1 + 1
POL( U13_1(x_1) ) = 2x_1 + 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U132_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 - 2}
POL( U133_4(x_1, ..., x_4) ) = x_2 + x_3 + 2
POL( U134_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = x_1 + x_2 + 2
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2x_1 + 1
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U62_1(x_1) ) = 2x_1 + 1
POL( U96_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = 2x_1 + 1
POL( U81_1(x_1) ) = x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(463)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(464) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2x_2 + 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = x_2 + 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U105_2(x_1, x_2) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( U132_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_3 + x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 + x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = x_3 + 2
POL( U136_4(x_1, ..., x_4) ) = x_1 + x_2 + 2x_3 + 2
POL( take_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U23_1(x_1) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = max{0, -2}
POL( U96_1(x_1) ) = 0
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 2
POL( MARK_1(x_1) ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(465)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(466) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(467)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(468) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2
POL( U105_2(x_1, x_2) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_1 - 1}
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = x_1 + 2
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 + 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 + x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, x_3 + x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( U31_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 2x_1 + 2
POL( U81_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)


----------------------------------------

(469)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(470) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(471)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(472) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = 0
   POL(U102(x_1, x_2, x_3)) = 0
   POL(U103(x_1, x_2, x_3)) = 0
   POL(U104(x_1, x_2, x_3)) = 0
   POL(U105(x_1, x_2)) = 0
   POL(U106(x_1)) = 0
   POL(U11(x_1, x_2)) = 0
   POL(U111(x_1, x_2, x_3)) = 0
   POL(U112(x_1, x_2, x_3)) = 0
   POL(U113(x_1, x_2, x_3)) = 0
   POL(U114(x_1, x_2)) = 0
   POL(U12(x_1, x_2)) = 0
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = 0
   POL(U13(x_1)) = 0
   POL(U131(x_1, x_2, x_3, x_4)) = 0
   POL(U132(x_1, x_2, x_3, x_4)) = 0
   POL(U133(x_1, x_2, x_3, x_4)) = 0
   POL(U134(x_1, x_2, x_3, x_4)) = 0
   POL(U135(x_1, x_2, x_3, x_4)) = 0
   POL(U136(x_1, x_2, x_3, x_4)) = 0
   POL(U21(x_1, x_2)) = x_1
   POL(U22(x_1, x_2)) = x_1
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = 0
   POL(U32(x_1, x_2)) = 0
   POL(U33(x_1)) = 0
   POL(U41(x_1, x_2, x_3)) = 0
   POL(U42(x_1, x_2, x_3)) = 0
   POL(U43(x_1, x_2, x_3)) = 0
   POL(U44(x_1, x_2, x_3)) = 0
   POL(U45(x_1, x_2)) = 0
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = 1 + x_1
   POL(U92(x_1, x_2, x_3)) = 1 + x_1
   POL(U93(x_1, x_2, x_3)) = 1 + x_1
   POL(U94(x_1, x_2, x_3)) = 1 + x_1
   POL(U95(x_1, x_2)) = 1 + x_1
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 1
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 0
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(473)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U61(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(474) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1 + 2
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 0
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U42_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U43_3(x_1, ..., x_3) ) = 2x_3 + 1
POL( U44_3(x_1, ..., x_3) ) = x_1 + 2x_3 + 1
POL( U45_2(x_1, x_2) ) = x_2 + 1
POL( U51_2(x_1, x_2) ) = 2x_1
POL( U61_2(x_1, x_2) ) = x_1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = x_1
POL( zeros ) = 0
POL( active_1(x_1) ) = x_1
POL( cons_2(x_1, x_2) ) = 2x_2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, -2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, -2}
POL( U105_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 1
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = x_1 + 1
POL( U11_2(x_1, x_2) ) = 2x_1 + 1
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( s_1(x_1) ) = x_1
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, -2}
POL( U122_1(x_1) ) = max{0, -2}
POL( U131_4(x_1, ..., x_4) ) = max{0, -2}
POL( U132_4(x_1, ..., x_4) ) = max{0, -2}
POL( U133_4(x_1, ..., x_4) ) = max{0, -2}
POL( U134_4(x_1, ..., x_4) ) = max{0, -2}
POL( U135_4(x_1, ..., x_4) ) = max{0, -2}
POL( U136_4(x_1, ..., x_4) ) = max{0, -2}
POL( take_2(x_1, x_2) ) = 0
POL( U23_1(x_1) ) = x_1
POL( U31_2(x_1, x_2) ) = 1
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = x_1
POL( U52_1(x_1) ) = x_1
POL( U62_1(x_1) ) = 2x_1
POL( U96_1(x_1) ) = x_1
POL( U71_1(x_1) ) = 2x_1
POL( U81_1(x_1) ) = x_1
POL( nil ) = 0
POL( MARK_1(x_1) ) = x_1 + 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(475)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U61(X1, X2)) -> MARK(X1)
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(476) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = 0
   POL(U102(x_1, x_2, x_3)) = 0
   POL(U103(x_1, x_2, x_3)) = 0
   POL(U104(x_1, x_2, x_3)) = 0
   POL(U105(x_1, x_2)) = 0
   POL(U106(x_1)) = 0
   POL(U11(x_1, x_2)) = 0
   POL(U111(x_1, x_2, x_3)) = x_2
   POL(U112(x_1, x_2, x_3)) = x_2
   POL(U113(x_1, x_2, x_3)) = x_2
   POL(U114(x_1, x_2)) = x_2
   POL(U12(x_1, x_2)) = 0
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = 0
   POL(U13(x_1)) = 0
   POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4
   POL(U21(x_1, x_2)) = 0
   POL(U22(x_1, x_2)) = 0
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = 0
   POL(U32(x_1, x_2)) = 0
   POL(U33(x_1)) = 0
   POL(U41(x_1, x_2, x_3)) = 0
   POL(U42(x_1, x_2, x_3)) = 0
   POL(U43(x_1, x_2, x_3)) = 0
   POL(U44(x_1, x_2, x_3)) = 0
   POL(U45(x_1, x_2)) = 0
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1 + x_2
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_1 + x_2
   POL(U62(x_1)) = x_1
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = 0
   POL(U92(x_1, x_2, x_3)) = 0
   POL(U93(x_1, x_2, x_3)) = 0
   POL(U94(x_1, x_2, x_3)) = 0
   POL(U95(x_1, x_2)) = 0
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = x_1
   POL(isNatKind(x_1)) = x_1
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 1 + x_1 + x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(477)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U61(X1, X2)) -> MARK(X1)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(478) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   MARK(U61(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = 0
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 0
POL( U42_3(x_1, ..., x_3) ) = 1
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2
POL( U105_2(x_1, x_2) ) = 2x_2 + 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( s_1(x_1) ) = 2x_1 + 1
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = x_2
POL( U122_1(x_1) ) = x_1 + 2
POL( U131_4(x_1, ..., x_4) ) = 2x_3 + 2x_4 + 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2
POL( U135_4(x_1, ..., x_4) ) = x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = x_1 + x_2 + 2
POL( take_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2x_1 + 1
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U62_1(x_1) ) = x_1 + 1
POL( U96_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(479)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(480) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 0
POL( U114_2(x_1, x_2) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 0
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 0
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( tt ) = 1
POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2x_2 + x_3
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( U11_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U12_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( s_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + 2x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 + x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U23_1(x_1) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, x_1 - 1}
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U61_2(x_1, x_2) ) = 2x_1 + 2
POL( U62_1(x_1) ) = 2
POL( U96_1(x_1) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, -2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(481)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U62(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(482) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U62(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 1
POL( U51_2(x_1, x_2) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U105_2(x_1, x_2) ) = 2x_2 + 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U12_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( s_1(x_1) ) = 2x_1 + 1
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U135_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2
POL( U136_4(x_1, ..., x_4) ) = 2x_3 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = max{0, -2}
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = x_1 + 1
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U62_1(x_1) ) = 2x_1 + 2
POL( U96_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = 2x_1 + 1
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(483)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(484) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = 0
   POL(U102(x_1, x_2, x_3)) = 0
   POL(U103(x_1, x_2, x_3)) = 0
   POL(U104(x_1, x_2, x_3)) = 0
   POL(U105(x_1, x_2)) = 0
   POL(U106(x_1)) = 0
   POL(U11(x_1, x_2)) = 0
   POL(U111(x_1, x_2, x_3)) = 1 + x_2
   POL(U112(x_1, x_2, x_3)) = 1 + x_2
   POL(U113(x_1, x_2, x_3)) = 1 + x_2
   POL(U114(x_1, x_2)) = 1 + x_2
   POL(U12(x_1, x_2)) = 0
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = 0
   POL(U13(x_1)) = 0
   POL(U131(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U133(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U134(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U135(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U136(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(U21(x_1, x_2)) = 0
   POL(U22(x_1, x_2)) = 0
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = 0
   POL(U32(x_1, x_2)) = 0
   POL(U33(x_1)) = 0
   POL(U41(x_1, x_2, x_3)) = 0
   POL(U42(x_1, x_2, x_3)) = 0
   POL(U43(x_1, x_2, x_3)) = 0
   POL(U44(x_1, x_2, x_3)) = 0
   POL(U45(x_1, x_2)) = 0
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1 + x_2
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_2
   POL(U62(x_1)) = 0
   POL(U71(x_1)) = x_1
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = 0
   POL(U92(x_1, x_2, x_3)) = 0
   POL(U93(x_1, x_2, x_3)) = 0
   POL(U94(x_1, x_2, x_3)) = 0
   POL(U95(x_1, x_2)) = 0
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = x_1
   POL(isNatKind(x_1)) = x_1
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 1 + x_1
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = x_2
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(485)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U71(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(486) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U71(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 1
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 0
POL( U51_2(x_1, x_2) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 1
POL( U92_3(x_1, ..., x_3) ) = 0
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = 0
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 1}
POL( tt ) = 1
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = x_1 + 2
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( s_1(x_1) ) = 2x_1 + 1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_3 + x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 + 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_3 + x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = 2x_1 + x_3 + x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = 2x_1 + 2
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = x_1 + 1
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = x_1 + 2
POL( U62_1(x_1) ) = 2
POL( U96_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = 2x_1 + 2
POL( U81_1(x_1) ) = x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(487)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> MARK(X1)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(488) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U51(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = 0
   POL(U102(x_1, x_2, x_3)) = 0
   POL(U103(x_1, x_2, x_3)) = 0
   POL(U104(x_1, x_2, x_3)) = 0
   POL(U105(x_1, x_2)) = 0
   POL(U106(x_1)) = 0
   POL(U11(x_1, x_2)) = 0
   POL(U111(x_1, x_2, x_3)) = 0
   POL(U112(x_1, x_2, x_3)) = 0
   POL(U113(x_1, x_2, x_3)) = 0
   POL(U114(x_1, x_2)) = 0
   POL(U12(x_1, x_2)) = 0
   POL(U121(x_1, x_2)) = 0
   POL(U122(x_1)) = 0
   POL(U13(x_1)) = 0
   POL(U131(x_1, x_2, x_3, x_4)) = 0
   POL(U132(x_1, x_2, x_3, x_4)) = 0
   POL(U133(x_1, x_2, x_3, x_4)) = 0
   POL(U134(x_1, x_2, x_3, x_4)) = 0
   POL(U135(x_1, x_2, x_3, x_4)) = 0
   POL(U136(x_1, x_2, x_3, x_4)) = 0
   POL(U21(x_1, x_2)) = 0
   POL(U22(x_1, x_2)) = 0
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = 0
   POL(U32(x_1, x_2)) = 0
   POL(U33(x_1)) = 0
   POL(U41(x_1, x_2, x_3)) = 0
   POL(U42(x_1, x_2, x_3)) = 0
   POL(U43(x_1, x_2, x_3)) = 0
   POL(U44(x_1, x_2, x_3)) = 0
   POL(U45(x_1, x_2)) = 0
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = 1 + x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = 0
   POL(U62(x_1)) = 0
   POL(U71(x_1)) = 0
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = 0
   POL(U92(x_1, x_2, x_3)) = 0
   POL(U93(x_1, x_2, x_3)) = 0
   POL(U94(x_1, x_2, x_3)) = 0
   POL(U95(x_1, x_2)) = 0
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 1
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 0
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(489)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(490) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = x_3
   POL(U102(x_1, x_2, x_3)) = x_2 + x_3
   POL(U103(x_1, x_2, x_3)) = x_2 + x_3
   POL(U104(x_1, x_2, x_3)) = x_2 + x_3
   POL(U105(x_1, x_2)) = x_2
   POL(U106(x_1)) = 0
   POL(U11(x_1, x_2)) = 0
   POL(U111(x_1, x_2, x_3)) = 0
   POL(U112(x_1, x_2, x_3)) = 0
   POL(U113(x_1, x_2, x_3)) = 0
   POL(U114(x_1, x_2)) = 0
   POL(U12(x_1, x_2)) = x_2
   POL(U121(x_1, x_2)) = x_2
   POL(U122(x_1)) = 0
   POL(U13(x_1)) = 0
   POL(U131(x_1, x_2, x_3, x_4)) = x_3 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(U133(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(U134(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(U135(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(U136(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(U21(x_1, x_2)) = 0
   POL(U22(x_1, x_2)) = 0
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_2
   POL(U32(x_1, x_2)) = x_2
   POL(U33(x_1)) = 0
   POL(U41(x_1, x_2, x_3)) = 1 + x_3
   POL(U42(x_1, x_2, x_3)) = 1 + x_3
   POL(U43(x_1, x_2, x_3)) = 1 + x_3
   POL(U44(x_1, x_2, x_3)) = 1 + x_3
   POL(U45(x_1, x_2)) = 1 + x_2
   POL(U46(x_1)) = x_1
   POL(U51(x_1, x_2)) = 1 + x_2
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2)) = x_2
   POL(U62(x_1)) = 0
   POL(U71(x_1)) = 0
   POL(U81(x_1)) = x_1
   POL(U91(x_1, x_2, x_3)) = 1 + x_3
   POL(U92(x_1, x_2, x_3)) = 1 + x_3
   POL(U93(x_1, x_2, x_3)) = 1 + x_3
   POL(U94(x_1, x_2, x_3)) = 1 + x_3
   POL(U95(x_1, x_2)) = 1 + x_2
   POL(U96(x_1)) = x_1
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = 1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 1 + x_1
   POL(isNatIListKind(x_1)) = 1 + x_1
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 1 + x_1
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(take(x_1, x_2)) = 0
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(491)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(492) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
----------------------------------------

(493)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(494) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = x_3 + 2
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 1
POL( U51_2(x_1, x_2) ) = 0
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 0
POL( U94_3(x_1, ..., x_3) ) = 1
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = x_3
POL( U105_2(x_1, x_2) ) = x_2 + 1
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( s_1(x_1) ) = 2x_1 + 1
POL( U13_1(x_1) ) = 2x_1 + 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2x_1 + 2
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_3 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, x_2 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_1 + x_2 + 2
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2
POL( U136_4(x_1, ..., x_4) ) = x_1 + 2
POL( take_2(x_1, x_2) ) = max{0, -2}
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U46_1(x_1) ) = x_1 + 1
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = x_1 + 2
POL( U96_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = 0
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(495)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(496) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = 0
POL( U45_2(x_1, x_2) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = 0
POL( U95_2(x_1, x_2) ) = 0
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = x_3 + 2
POL( U105_2(x_1, x_2) ) = x_1
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = 2x_2 + 2
POL( U12_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( s_1(x_1) ) = 2
POL( U13_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, x_3 + 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2x_3 + x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2x_3 + 2
POL( U136_4(x_1, ..., x_4) ) = x_1 + x_2 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = x_1 + 1
POL( U23_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 1}
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = x_1
POL( U62_1(x_1) ) = 2
POL( U96_1(x_1) ) = 0
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)


----------------------------------------

(497)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(498) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 1
POL( U94_3(x_1, ..., x_3) ) = 0
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + 2x_3 + 2
POL( U105_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = x_1
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( s_1(x_1) ) = x_1 + 1
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2x_3 + 2x_4 + 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_3 + 2
POL( U135_4(x_1, ..., x_4) ) = x_1 + 2x_2 + x_3 + x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, x_3 + x_4 - 2}
POL( take_2(x_1, x_2) ) = x_2 + 2
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U46_1(x_1) ) = x_1 + 1
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 2
POL( U62_1(x_1) ) = 2
POL( U96_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(499)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(500) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = 1
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 0
POL( U43_3(x_1, ..., x_3) ) = 0
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, 2x_1 - 1}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( isNatIListKind_1(x_1) ) = x_1 + 2
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 2x_1
POL( isNatIList_1(x_1) ) = x_1
POL( U11_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = 0
POL( U122_1(x_1) ) = max{0, -2}
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_3 + 2x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_1 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U23_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U61_2(x_1, x_2) ) = x_1 + 2
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = x_2 + 2
POL( U96_1(x_1) ) = x_1
POL( U71_1(x_1) ) = 2x_1 + 1
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, -2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(501)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(502) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 1
POL( U44_3(x_1, ..., x_3) ) = 0
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = x_3 + 2
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = 2
POL( zeros ) = 1
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2x_1 + 2
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 1}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 1
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( s_1(x_1) ) = 2x_1 + 1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U122_1(x_1) ) = x_1 + 2
POL( U131_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 2
POL( U132_4(x_1, ..., x_4) ) = max{0, x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2
POL( U136_4(x_1, ..., x_4) ) = x_1 + x_3 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = max{0, -2}
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = 2x_2 + 2
POL( U32_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = x_1 + 2
POL( U46_1(x_1) ) = x_1 + 1
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U62_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = x_2 + 2
POL( U96_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(503)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(504) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = x_1
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( s_1(x_1) ) = 2
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = x_2 + 2
POL( U122_1(x_1) ) = x_1
POL( U131_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 2
POL( U132_4(x_1, ..., x_4) ) = x_2 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = 2x_3 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = 2x_1 + 2
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = 1
POL( U41_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( U51_2(x_1, x_2) ) = 2x_1 + 2
POL( U52_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U96_1(x_1) ) = 2
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(505)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(506) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U52(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U12_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( s_1(x_1) ) = x_1 + 1
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2
POL( U132_4(x_1, ..., x_4) ) = x_1 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, x_2 - 2}
POL( take_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U46_1(x_1) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U52_1(x_1) ) = x_1 + 2
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U96_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 2
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(507)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(508) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 0
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 1
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( tt ) = 1
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U105_2(x_1, x_2) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 2x_1
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( s_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = 2x_1
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 1
POL( U131_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 + x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = 2x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U23_1(x_1) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( U51_2(x_1, x_2) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = max{0, -2}
POL( U62_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( U92_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 + 2
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = max{0, -2}
POL( nil ) = 1
POL( MARK_1(x_1) ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(509)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(510) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = 0
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = 0
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U105_2(x_1, x_2) ) = 0
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 1
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2x_2 + 2
POL( s_1(x_1) ) = 2x_1 + 1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U122_1(x_1) ) = 0
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_3 + 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_1 + 2x_3 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_3 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U41_3(x_1, ..., x_3) ) = x_1 + x_3
POL( U42_3(x_1, ..., x_3) ) = x_3 + 2
POL( U46_1(x_1) ) = x_1 + 1
POL( U51_2(x_1, x_2) ) = x_1 + 2x_2 + 1
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U92_3(x_1, ..., x_3) ) = x_1
POL( U96_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(511)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(512) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 0
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = 1
POL( U94_3(x_1, ..., x_3) ) = 1
POL( U95_2(x_1, x_2) ) = 1
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = x_1 + x_3 + 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U105_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U12_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( s_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, x_2 - 2}
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U23_1(x_1) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U62_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( U81_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, -2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(513)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(514) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 2
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 2x_1 + 1
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = x_3 + 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( s_1(x_1) ) = 2x_1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, x_2 - 1}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 + x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_3 + x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_3 + x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = x_1
POL( U31_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 1}
POL( U33_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U46_1(x_1) ) = x_1
POL( U51_2(x_1, x_2) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U96_1(x_1) ) = 2x_1
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = x_1
POL( nil ) = 0
POL( MARK_1(x_1) ) = 2x_1 + 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(515)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(516) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 1}
POL( U12_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U122_1(x_1) ) = 1
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 + x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = 2x_1 + 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( U43_3(x_1, ..., x_3) ) = x_1 + x_2 + 2
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U62_1(x_1) ) = 0
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U96_1(x_1) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U81_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)


----------------------------------------

(517)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(518) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( U44_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U45_2(x_1, x_2) ) = x_1 + 1
POL( U95_2(x_1, x_2) ) = 2x_2 + 2
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = x_3 + 2
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 1}
POL( s_1(x_1) ) = 2x_1 + 1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U122_1(x_1) ) = x_1
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 + 2x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_2 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U46_1(x_1) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U52_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U92_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U93_3(x_1, ..., x_3) ) = x_2 + 2
POL( U94_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U96_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U81_1(x_1) ) = x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(519)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(520) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = 1
POL( U111_3(x_1, ..., x_3) ) = x_1 + 1
POL( U112_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U114_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( U45_2(x_1, x_2) ) = 1
POL( length_1(x_1) ) = 1
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( tt ) = 1
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNatKind_1(x_1) ) = 1
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 1
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( s_1(x_1) ) = x_1
POL( U13_1(x_1) ) = x_1 + 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = x_1
POL( U131_4(x_1, ..., x_4) ) = x_1 + 2x_4 + 2
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, x_1 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( take_2(x_1, x_2) ) = 0
POL( U23_1(x_1) ) = x_1
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U42_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U46_1(x_1) ) = x_1
POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U94_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U95_2(x_1, x_2) ) = 2
POL( U96_1(x_1) ) = 2x_1
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = x_1
POL( nil ) = 0
POL( MARK_1(x_1) ) = x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(521)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(522) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U46(X)) -> MARK(X)
   MARK(U96(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = x_1 + x_3 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2x_2 + 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U12_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( s_1(x_1) ) = x_1 + 1
POL( U13_1(x_1) ) = x_1 + 1
POL( isNatList_1(x_1) ) = 1
POL( U121_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U122_1(x_1) ) = x_1 + 1
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_1 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_2 + 2
POL( U135_4(x_1, ..., x_4) ) = x_1 + x_2 + 2x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = x_3 + 2
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U41_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( U46_1(x_1) ) = 2x_1 + 2
POL( U51_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U52_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = x_2 + 2
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( U93_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 1
POL( U95_2(x_1, x_2) ) = x_1 + 2
POL( U96_1(x_1) ) = x_1 + 2
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(523)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(524) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = 1
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 1
POL( mark_1(x_1) ) = 0
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = x_1 + x_2 + 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( s_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 2x_1
POL( U121_2(x_1, x_2) ) = x_2 + 2
POL( U122_1(x_1) ) = 0
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 + x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 + 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = x_2 + 2
POL( take_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( U31_2(x_1, x_2) ) = 2x_2 + 2
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U33_1(x_1) ) = 1
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U52_1(x_1) ) = 1
POL( U61_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U62_1(x_1) ) = x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 1
POL( U92_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( U93_3(x_1, ..., x_3) ) = x_2 + 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U96_1(x_1) ) = 2
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, -2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)


----------------------------------------

(525)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(526) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 + 2
POL( U105_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( s_1(x_1) ) = x_1 + 1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = x_2 + 2x_3 + x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 + 2x_4 - 1}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = max{0, -2}
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 1}
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = max{0, x_1 - 1}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( U43_3(x_1, ..., x_3) ) = x_1 + 2
POL( U44_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( U46_1(x_1) ) = 0
POL( U51_2(x_1, x_2) ) = 2
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2
POL( U95_2(x_1, x_2) ) = 2
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(527)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(528) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = 0
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( isNatKind_1(x_1) ) = 2x_1 + 2
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U12_2(x_1, x_2) ) = 1
POL( s_1(x_1) ) = x_1 + 2
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U122_1(x_1) ) = 1
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 - 1}
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + 2x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_1 + x_2 + x_4
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2
POL( U136_4(x_1, ..., x_4) ) = x_1 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = 2x_1 + 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U45_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U46_1(x_1) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U62_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( U92_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 1}
POL( U93_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U95_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U96_1(x_1) ) = 1
POL( U71_1(x_1) ) = x_1
POL( U81_1(x_1) ) = x_1 + 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)


----------------------------------------

(529)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U81(X)) -> MARK(X)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(530) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(531)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
   MARK(U81(X)) -> MARK(X)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(532) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U81(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( zeros ) = 2
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = x_1
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2x_1 + 2
POL( s_1(x_1) ) = x_1 + 1
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 + x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, x_3 + 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U46_1(x_1) ) = x_1
POL( U51_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U52_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U62_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U93_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U95_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U96_1(x_1) ) = x_1
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U81_1(x_1) ) = x_1 + 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(533)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(534) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(MARK(x_1)) =  	[[0]] 	 +  	[[1, 0]] 	* 	x_1
>>>

   <<<
 POL(isNat(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[0]] 	 +  	[[1, 0]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U111(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[1], [1]]
>>>

   <<<
 POL(U112(x_1, x_2, x_3)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(isNatIListKind(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(U113(x_1, x_2, x_3)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U114(x_1, x_2)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(length(x_1)) =  	[[1], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U23(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(isNatList(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(zeros) =  	[[0], [1]]
>>>

   <<<
 POL(active(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[1], [1]]
>>>

   <<<
 POL(U101(x_1, x_2, x_3)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U102(x_1, x_2, x_3)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U103(x_1, x_2, x_3)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U104(x_1, x_2, x_3)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U105(x_1, x_2)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U106(x_1)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[1], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U12(x_1, x_2)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U121(x_1, x_2)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U122(x_1)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U131(x_1, x_2, x_3, x_4)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [1, 1]] 	* 	x_3 	 +  	[[0, 0], [0, 0]] 	* 	x_4
>>>

   <<<
 POL(U132(x_1, x_2, x_3, x_4)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [1, 1]] 	* 	x_3 	 +  	[[0, 0], [0, 0]] 	* 	x_4
>>>

   <<<
 POL(U133(x_1, x_2, x_3, x_4)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 1], [0, 0]] 	* 	x_3 	 +  	[[0, 0], [0, 0]] 	* 	x_4
>>>

   <<<
 POL(U134(x_1, x_2, x_3, x_4)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [1, 1]] 	* 	x_3 	 +  	[[0, 0], [0, 0]] 	* 	x_4
>>>

   <<<
 POL(U135(x_1, x_2, x_3, x_4)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [1, 1]] 	* 	x_3 	 +  	[[0, 0], [0, 0]] 	* 	x_4
>>>

   <<<
 POL(U136(x_1, x_2, x_3, x_4)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 1], [0, 0]] 	* 	x_3 	 +  	[[0, 0], [0, 0]] 	* 	x_4
>>>

   <<<
 POL(take(x_1, x_2)) =  	[[1], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U31(x_1, x_2)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U32(x_1, x_2)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(U33(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U41(x_1, x_2, x_3)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 1], [1, 0]] 	* 	x_3
>>>

   <<<
 POL(U42(x_1, x_2, x_3)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 1], [1, 0]] 	* 	x_3
>>>

   <<<
 POL(U43(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U44(x_1, x_2, x_3)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U45(x_1, x_2)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U46(x_1)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(U52(x_1)) =  	[[0], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U91(x_1, x_2, x_3)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U92(x_1, x_2, x_3)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [1, 0]] 	* 	x_3
>>>

   <<<
 POL(U93(x_1, x_2, x_3)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [1, 0]] 	* 	x_3
>>>

   <<<
 POL(U94(x_1, x_2, x_3)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[0, 0], [1, 0]] 	* 	x_3
>>>

   <<<
 POL(U95(x_1, x_2)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(U96(x_1)) =  	[[0], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1)) =  	[[0], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(U81(x_1)) =  	[[0], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(nil) =  	[[1], [0]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)


----------------------------------------

(535)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(536) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 1}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U114_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( length_1(x_1) ) = 1
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = 2x_2 + 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( isNatIListKind_1(x_1) ) = x_1
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = 1
POL( U106_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U12_2(x_1, x_2) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = x_1
POL( U121_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U122_1(x_1) ) = x_1 + 1
POL( U131_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 + 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_2 + x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = x_1 + x_2 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 - 2}
POL( U136_4(x_1, ..., x_4) ) = 2x_3 + 2
POL( take_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( U31_2(x_1, x_2) ) = x_2 + 1
POL( U32_2(x_1, x_2) ) = 2x_1 + 2
POL( U33_1(x_1) ) = 1
POL( U41_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U46_1(x_1) ) = 2
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 + 2
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U95_2(x_1, x_2) ) = 2x_1 + 2
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U81_1(x_1) ) = 2x_1 + 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)


----------------------------------------

(537)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(538) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(539)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(540) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U114_2(x_1, x_2) ) = 2x_1 + 1
POL( U21_2(x_1, x_2) ) = 2x_1 + 1
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U105_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U12_2(x_1, x_2) ) = 2x_2 + 2
POL( U111_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( s_1(x_1) ) = x_1 + 1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U122_1(x_1) ) = max{0, -2}
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U23_1(x_1) ) = x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = x_3 + 2
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( U51_2(x_1, x_2) ) = x_1
POL( U52_1(x_1) ) = 0
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = x_3 + 2
POL( U92_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U93_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U96_1(x_1) ) = 2
POL( U71_1(x_1) ) = 0
POL( U81_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(541)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(542) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2
POL( U105_2(x_1, x_2) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = x_1
POL( U11_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U12_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = x_1 + 2
POL( U121_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 + x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, -2}
POL( U23_1(x_1) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U43_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( U51_2(x_1, x_2) ) = 2
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U61_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U62_1(x_1) ) = 2x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U95_2(x_1, x_2) ) = 2x_1 + 2
POL( U96_1(x_1) ) = 2x_1 + 2
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, -2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)


----------------------------------------

(543)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(544) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(length(X)) -> ACTIVE(length(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 0
POL( mark_1(x_1) ) = max{0, -2}
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = x_1 + x_2 + 2x_3 + 2
POL( isNatIListKind_1(x_1) ) = 2x_1 + 2
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( U105_2(x_1, x_2) ) = x_2 + 2
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 2x_1
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U112_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2}
POL( s_1(x_1) ) = x_1
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 2x_1
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U23_1(x_1) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U33_1(x_1) ) = max{0, x_1 - 2}
POL( U41_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U43_3(x_1, ..., x_3) ) = x_1 + 2
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U45_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U46_1(x_1) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U52_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2}
POL( U92_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 2
POL( U93_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2
POL( U94_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U95_2(x_1, x_2) ) = 2
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = 2
POL( nil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)


----------------------------------------

(545)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(546) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( U114_2(x_1, x_2) ) = 1
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( mark_1(x_1) ) = 2
POL( zeros ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2x_2 + 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 1}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = x_1 + 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = x_1 + 2
POL( U11_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U111_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( U112_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2
POL( s_1(x_1) ) = 2x_1 + 1
POL( length_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = x_1 + 2
POL( U121_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + x_3 + 2x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 + x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2
POL( take_2(x_1, x_2) ) = 1
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = 0
POL( U41_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U43_3(x_1, ..., x_3) ) = x_1 + x_2 + 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2x_1 + 2
POL( U46_1(x_1) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U52_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2
POL( U93_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U94_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( U95_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U96_1(x_1) ) = 2
POL( U71_1(x_1) ) = max{0, 2x_1 - 1}
POL( U81_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 1
POL( MARK_1(x_1) ) = max{0, x_1 - 1}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none


----------------------------------------

(547)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(548) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = x_1
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( mark_1(x_1) ) = 2
POL( zeros ) = 1
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 1}
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2x_1 + 2
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 1}
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U113_3(x_1, ..., x_3) ) = 1
POL( s_1(x_1) ) = 0
POL( length_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = 2x_1 + 2x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = 2x_2 + x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = 0
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( U31_2(x_1, x_2) ) = 2x_2 + 2
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, x_1 - 2}
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( U45_2(x_1, x_2) ) = 2
POL( U46_1(x_1) ) = 2
POL( U51_2(x_1, x_2) ) = 2x_1 + x_2
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U92_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U93_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U94_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( U95_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U96_1(x_1) ) = 2x_1 + 2
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = x_1 + 1
POL( nil ) = 0
POL( MARK_1(x_1) ) = max{0, -2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)


----------------------------------------

(549)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(s(X)) -> MARK(X)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U23(X)) -> MARK(X)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(550) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(s(X)) -> MARK(X)
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = x_1
   POL(MARK(x_1)) = x_1
   POL(U101(x_1, x_2, x_3)) = x_3
   POL(U102(x_1, x_2, x_3)) = x_2 + x_3
   POL(U103(x_1, x_2, x_3)) = x_2 + x_3
   POL(U104(x_1, x_2, x_3)) = x_2 + x_3
   POL(U105(x_1, x_2)) = x_2
   POL(U106(x_1)) = 0
   POL(U11(x_1, x_2)) = 0
   POL(U111(x_1, x_2, x_3)) = x_3
   POL(U112(x_1, x_2, x_3)) = x_3
   POL(U113(x_1, x_2, x_3)) = x_2 + x_3
   POL(U114(x_1, x_2)) = 1
   POL(U12(x_1, x_2)) = x_2
   POL(U121(x_1, x_2)) = x_2
   POL(U122(x_1)) = 0
   POL(U13(x_1)) = 0
   POL(U131(x_1, x_2, x_3, x_4)) = x_3 + x_4
   POL(U132(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(U133(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(U134(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(U135(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(U136(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(U21(x_1, x_2)) = 1 + x_2
   POL(U22(x_1, x_2)) = 1 + x_2
   POL(U23(x_1)) = x_1
   POL(U31(x_1, x_2)) = x_2
   POL(U32(x_1, x_2)) = x_2
   POL(U33(x_1)) = 0
   POL(U41(x_1, x_2, x_3)) = x_3
   POL(U42(x_1, x_2, x_3)) = x_2 + x_3
   POL(U43(x_1, x_2, x_3)) = x_2 + x_3
   POL(U44(x_1, x_2, x_3)) = x_2 + x_3
   POL(U45(x_1, x_2)) = x_2
   POL(U46(x_1)) = 0
   POL(U51(x_1, x_2)) = x_2
   POL(U52(x_1)) = 0
   POL(U61(x_1, x_2)) = x_2
   POL(U62(x_1)) = 0
   POL(U71(x_1)) = 0
   POL(U81(x_1)) = 0
   POL(U91(x_1, x_2, x_3)) = x_3
   POL(U92(x_1, x_2, x_3)) = x_2 + x_3
   POL(U93(x_1, x_2, x_3)) = x_2 + x_3
   POL(U94(x_1, x_2, x_3)) = x_2 + x_3
   POL(U95(x_1, x_2)) = x_2
   POL(U96(x_1)) = 0
   POL(active(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(isNat(x_1)) = x_1
   POL(isNatIList(x_1)) = 0
   POL(isNatIListKind(x_1)) = 0
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 0
   POL(mark(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = 1 + x_1
   POL(take(x_1, x_2)) = 0
   POL(tt) = 0
   POL(zeros) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)


----------------------------------------

(551)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(552) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes.
----------------------------------------

(553)
Complex Obligation (AND)

----------------------------------------

(554)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(555) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2)) we obtained the following new rules [LPAR04]:

   (MARK(U22(isNatKind(z0), z0)) -> ACTIVE(U22(mark(isNatKind(z0)), z0)),MARK(U22(isNatKind(z0), z0)) -> ACTIVE(U22(mark(isNatKind(z0)), z0)))


----------------------------------------

(556)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(isNatKind(z0), z0)) -> ACTIVE(U22(mark(isNatKind(z0)), z0))

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(557) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(U22(isNatKind(z0), z0)) -> ACTIVE(U22(mark(isNatKind(z0)), z0))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
ACTIVE(x1)  =  x1

U21(x1, x2)  =  U21

MARK(x1)  =  MARK

U22(x1, x2)  =  U22


Knuth-Bendix order [KBO] with precedence:MARK > U22

and weight map:

   U21=2
   MARK=1
   U22=1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)


----------------------------------------

(558)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(559) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(560)
YES

----------------------------------------

(561)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U23(X)) -> MARK(X)

The TRS R consists of the following rules:

   mark(zeros) -> active(zeros)
   mark(cons(X1, X2)) -> active(cons(mark(X1), X2))
   active(zeros) -> mark(cons(0, zeros))
   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(s(X)) -> active(s(mark(X)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U33(X)) -> active(U33(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U52(X)) -> active(U52(mark(X)))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(U62(X)) -> active(U62(mark(X)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U92(X1, X2, X3)) -> active(U92(mark(X1), X2, X3))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U93(X1, X2, X3)) -> active(U93(mark(X1), X2, X3))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U94(X1, X2, X3)) -> active(U94(mark(X1), X2, X3))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U95(X1, X2)) -> active(U95(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U96(X)) -> active(U96(mark(X)))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U71(X)) -> active(U71(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U81(X)) -> active(U81(mark(X)))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U91(X1, X2, X3)) -> active(U91(mark(X1), X2, X3))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U111(X1, X2, X3)) -> active(U111(mark(X1), X2, X3))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(isNatKind(X)) -> active(isNatKind(X))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   mark(isNat(X)) -> active(isNat(X))
   mark(isNatIList(X)) -> active(isNatIList(X))
   mark(length(X)) -> active(length(mark(X)))
   mark(isNatList(X)) -> active(isNatList(X))
   mark(nil) -> active(nil)
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)
   take(X1, mark(X2)) -> take(X1, X2)
   take(mark(X1), X2) -> take(X1, X2)
   take(active(X1), X2) -> take(X1, X2)
   take(X1, active(X2)) -> take(X1, X2)
   active(U106(tt)) -> mark(tt)
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U23(tt)) -> mark(tt)
   active(U33(tt)) -> mark(tt)
   active(U46(tt)) -> mark(tt)
   active(U52(tt)) -> mark(tt)
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U96(tt)) -> mark(tt)
   active(isNat(0)) -> mark(tt)
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatKind(0)) -> mark(tt)
   active(isNatList(nil)) -> mark(tt)
   active(length(nil)) -> mark(0)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   U102(X1, mark(X2), X3) -> U102(X1, X2, X3)
   U102(mark(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, X2, mark(X3)) -> U102(X1, X2, X3)
   U102(active(X1), X2, X3) -> U102(X1, X2, X3)
   U102(X1, active(X2), X3) -> U102(X1, X2, X3)
   U102(X1, X2, active(X3)) -> U102(X1, X2, X3)
   U103(X1, mark(X2), X3) -> U103(X1, X2, X3)
   U103(mark(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, X2, mark(X3)) -> U103(X1, X2, X3)
   U103(active(X1), X2, X3) -> U103(X1, X2, X3)
   U103(X1, active(X2), X3) -> U103(X1, X2, X3)
   U103(X1, X2, active(X3)) -> U103(X1, X2, X3)
   U104(X1, mark(X2), X3) -> U104(X1, X2, X3)
   U104(mark(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, X2, mark(X3)) -> U104(X1, X2, X3)
   U104(active(X1), X2, X3) -> U104(X1, X2, X3)
   U104(X1, active(X2), X3) -> U104(X1, X2, X3)
   U104(X1, X2, active(X3)) -> U104(X1, X2, X3)
   U105(X1, mark(X2)) -> U105(X1, X2)
   U105(mark(X1), X2) -> U105(X1, X2)
   U105(active(X1), X2) -> U105(X1, X2)
   U105(X1, active(X2)) -> U105(X1, X2)
   U106(active(X)) -> U106(X)
   U106(mark(X)) -> U106(X)
   U12(X1, mark(X2)) -> U12(X1, X2)
   U12(mark(X1), X2) -> U12(X1, X2)
   U12(active(X1), X2) -> U12(X1, X2)
   U12(X1, active(X2)) -> U12(X1, X2)
   U112(X1, mark(X2), X3) -> U112(X1, X2, X3)
   U112(mark(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, X2, mark(X3)) -> U112(X1, X2, X3)
   U112(active(X1), X2, X3) -> U112(X1, X2, X3)
   U112(X1, active(X2), X3) -> U112(X1, X2, X3)
   U112(X1, X2, active(X3)) -> U112(X1, X2, X3)
   U113(X1, mark(X2), X3) -> U113(X1, X2, X3)
   U113(mark(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, X2, mark(X3)) -> U113(X1, X2, X3)
   U113(active(X1), X2, X3) -> U113(X1, X2, X3)
   U113(X1, active(X2), X3) -> U113(X1, X2, X3)
   U113(X1, X2, active(X3)) -> U113(X1, X2, X3)
   U114(X1, mark(X2)) -> U114(X1, X2)
   U114(mark(X1), X2) -> U114(X1, X2)
   U114(active(X1), X2) -> U114(X1, X2)
   U114(X1, active(X2)) -> U114(X1, X2)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(X)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(X)
   U132(X1, mark(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(mark(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, mark(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, mark(X4)) -> U132(X1, X2, X3, X4)
   U132(active(X1), X2, X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, active(X2), X3, X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, active(X3), X4) -> U132(X1, X2, X3, X4)
   U132(X1, X2, X3, active(X4)) -> U132(X1, X2, X3, X4)
   U133(X1, mark(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(mark(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, mark(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, mark(X4)) -> U133(X1, X2, X3, X4)
   U133(active(X1), X2, X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, active(X2), X3, X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, active(X3), X4) -> U133(X1, X2, X3, X4)
   U133(X1, X2, X3, active(X4)) -> U133(X1, X2, X3, X4)
   U134(X1, mark(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(mark(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, mark(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, mark(X4)) -> U134(X1, X2, X3, X4)
   U134(active(X1), X2, X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, active(X2), X3, X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, active(X3), X4) -> U134(X1, X2, X3, X4)
   U134(X1, X2, X3, active(X4)) -> U134(X1, X2, X3, X4)
   U135(X1, mark(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(mark(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, mark(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, mark(X4)) -> U135(X1, X2, X3, X4)
   U135(active(X1), X2, X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, active(X2), X3, X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, active(X3), X4) -> U135(X1, X2, X3, X4)
   U135(X1, X2, X3, active(X4)) -> U135(X1, X2, X3, X4)
   U136(X1, mark(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(mark(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, mark(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, mark(X4)) -> U136(X1, X2, X3, X4)
   U136(active(X1), X2, X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, active(X2), X3, X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, active(X3), X4) -> U136(X1, X2, X3, X4)
   U136(X1, X2, X3, active(X4)) -> U136(X1, X2, X3, X4)
   U22(X1, mark(X2)) -> U22(X1, X2)
   U22(mark(X1), X2) -> U22(X1, X2)
   U22(active(X1), X2) -> U22(X1, X2)
   U22(X1, active(X2)) -> U22(X1, X2)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   U32(X1, mark(X2)) -> U32(X1, X2)
   U32(mark(X1), X2) -> U32(X1, X2)
   U32(active(X1), X2) -> U32(X1, X2)
   U32(X1, active(X2)) -> U32(X1, X2)
   U33(active(X)) -> U33(X)
   U33(mark(X)) -> U33(X)
   U42(X1, mark(X2), X3) -> U42(X1, X2, X3)
   U42(mark(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, X2, mark(X3)) -> U42(X1, X2, X3)
   U42(active(X1), X2, X3) -> U42(X1, X2, X3)
   U42(X1, active(X2), X3) -> U42(X1, X2, X3)
   U42(X1, X2, active(X3)) -> U42(X1, X2, X3)
   U43(X1, mark(X2), X3) -> U43(X1, X2, X3)
   U43(mark(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, X2, mark(X3)) -> U43(X1, X2, X3)
   U43(active(X1), X2, X3) -> U43(X1, X2, X3)
   U43(X1, active(X2), X3) -> U43(X1, X2, X3)
   U43(X1, X2, active(X3)) -> U43(X1, X2, X3)
   U44(X1, mark(X2), X3) -> U44(X1, X2, X3)
   U44(mark(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, X2, mark(X3)) -> U44(X1, X2, X3)
   U44(active(X1), X2, X3) -> U44(X1, X2, X3)
   U44(X1, active(X2), X3) -> U44(X1, X2, X3)
   U44(X1, X2, active(X3)) -> U44(X1, X2, X3)
   U45(X1, mark(X2)) -> U45(X1, X2)
   U45(mark(X1), X2) -> U45(X1, X2)
   U45(active(X1), X2) -> U45(X1, X2)
   U45(X1, active(X2)) -> U45(X1, X2)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   U92(X1, mark(X2), X3) -> U92(X1, X2, X3)
   U92(mark(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, X2, mark(X3)) -> U92(X1, X2, X3)
   U92(active(X1), X2, X3) -> U92(X1, X2, X3)
   U92(X1, active(X2), X3) -> U92(X1, X2, X3)
   U92(X1, X2, active(X3)) -> U92(X1, X2, X3)
   U93(X1, mark(X2), X3) -> U93(X1, X2, X3)
   U93(mark(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, X2, mark(X3)) -> U93(X1, X2, X3)
   U93(active(X1), X2, X3) -> U93(X1, X2, X3)
   U93(X1, active(X2), X3) -> U93(X1, X2, X3)
   U93(X1, X2, active(X3)) -> U93(X1, X2, X3)
   U94(X1, mark(X2), X3) -> U94(X1, X2, X3)
   U94(mark(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, X2, mark(X3)) -> U94(X1, X2, X3)
   U94(active(X1), X2, X3) -> U94(X1, X2, X3)
   U94(X1, active(X2), X3) -> U94(X1, X2, X3)
   U94(X1, X2, active(X3)) -> U94(X1, X2, X3)
   U95(X1, mark(X2)) -> U95(X1, X2)
   U95(mark(X1), X2) -> U95(X1, X2)
   U95(active(X1), X2) -> U95(X1, X2)
   U95(X1, active(X2)) -> U95(X1, X2)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(X1, mark(X2)) -> U21(X1, X2)
   U21(mark(X1), X2) -> U21(X1, X2)
   U21(active(X1), X2) -> U21(X1, X2)
   U21(X1, active(X2)) -> U21(X1, X2)
   U31(X1, mark(X2)) -> U31(X1, X2)
   U31(mark(X1), X2) -> U31(X1, X2)
   U31(active(X1), X2) -> U31(X1, X2)
   U31(X1, active(X2)) -> U31(X1, X2)
   U41(X1, mark(X2), X3) -> U41(X1, X2, X3)
   U41(mark(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, X2, mark(X3)) -> U41(X1, X2, X3)
   U41(active(X1), X2, X3) -> U41(X1, X2, X3)
   U41(X1, active(X2), X3) -> U41(X1, X2, X3)
   U41(X1, X2, active(X3)) -> U41(X1, X2, X3)
   U51(X1, mark(X2)) -> U51(X1, X2)
   U51(mark(X1), X2) -> U51(X1, X2)
   U51(active(X1), X2) -> U51(X1, X2)
   U51(X1, active(X2)) -> U51(X1, X2)
   U61(X1, mark(X2)) -> U61(X1, X2)
   U61(mark(X1), X2) -> U61(X1, X2)
   U61(active(X1), X2) -> U61(X1, X2)
   U61(X1, active(X2)) -> U61(X1, X2)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U91(X1, mark(X2), X3) -> U91(X1, X2, X3)
   U91(mark(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, X2, mark(X3)) -> U91(X1, X2, X3)
   U91(active(X1), X2, X3) -> U91(X1, X2, X3)
   U91(X1, active(X2), X3) -> U91(X1, X2, X3)
   U91(X1, X2, active(X3)) -> U91(X1, X2, X3)
   U101(X1, mark(X2), X3) -> U101(X1, X2, X3)
   U101(mark(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, X2, mark(X3)) -> U101(X1, X2, X3)
   U101(active(X1), X2, X3) -> U101(X1, X2, X3)
   U101(X1, active(X2), X3) -> U101(X1, X2, X3)
   U101(X1, X2, active(X3)) -> U101(X1, X2, X3)
   U111(X1, mark(X2), X3) -> U111(X1, X2, X3)
   U111(mark(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, X2, mark(X3)) -> U111(X1, X2, X3)
   U111(active(X1), X2, X3) -> U111(X1, X2, X3)
   U111(X1, active(X2), X3) -> U111(X1, X2, X3)
   U111(X1, X2, active(X3)) -> U111(X1, X2, X3)
   U121(X1, mark(X2)) -> U121(X1, X2)
   U121(mark(X1), X2) -> U121(X1, X2)
   U121(active(X1), X2) -> U121(X1, X2)
   U121(X1, active(X2)) -> U121(X1, X2)
   U131(X1, mark(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(mark(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, mark(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, mark(X4)) -> U131(X1, X2, X3, X4)
   U131(active(X1), X2, X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, active(X2), X3, X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, active(X3), X4) -> U131(X1, X2, X3, X4)
   U131(X1, X2, X3, active(X4)) -> U131(X1, X2, X3, X4)

The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(562) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(563)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U23(X)) -> MARK(X)

R is empty.
The set Q consists of the following terms:

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U23(mark(x0))
   U23(active(x0))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(564) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   active(zeros)
   active(U101(tt, x0, x1))
   active(U102(tt, x0, x1))
   active(U103(tt, x0, x1))
   active(U104(tt, x0, x1))
   active(U105(tt, x0))
   active(U106(tt))
   active(U11(tt, x0))
   active(U111(tt, x0, x1))
   active(U112(tt, x0, x1))
   active(U113(tt, x0, x1))
   active(U114(tt, x0))
   active(U12(tt, x0))
   active(U121(tt, x0))
   active(U122(tt))
   active(U13(tt))
   active(U131(tt, x0, x1, x2))
   active(U132(tt, x0, x1, x2))
   active(U133(tt, x0, x1, x2))
   active(U134(tt, x0, x1, x2))
   active(U135(tt, x0, x1, x2))
   active(U136(tt, x0, x1, x2))
   active(U21(tt, x0))
   active(U22(tt, x0))
   active(U23(tt))
   active(U31(tt, x0))
   active(U32(tt, x0))
   active(U33(tt))
   active(U41(tt, x0, x1))
   active(U42(tt, x0, x1))
   active(U43(tt, x0, x1))
   active(U44(tt, x0, x1))
   active(U45(tt, x0))
   active(U46(tt))
   active(U51(tt, x0))
   active(U52(tt))
   active(U61(tt, x0))
   active(U62(tt))
   active(U71(tt))
   active(U81(tt))
   active(U91(tt, x0, x1))
   active(U92(tt, x0, x1))
   active(U93(tt, x0, x1))
   active(U94(tt, x0, x1))
   active(U95(tt, x0))
   active(U96(tt))
   active(isNat(0))
   active(isNat(length(x0)))
   active(isNat(s(x0)))
   active(isNatIList(x0))
   active(isNatIListKind(nil))
   active(isNatIListKind(zeros))
   active(isNatIListKind(cons(x0, x1)))
   active(isNatIListKind(take(x0, x1)))
   active(isNatKind(0))
   active(isNatKind(length(x0)))
   active(isNatKind(s(x0)))
   active(isNatList(nil))
   active(isNatList(cons(x0, x1)))
   active(isNatList(take(x0, x1)))
   active(length(nil))
   active(length(cons(x0, x1)))
   active(take(0, x0))
   active(take(s(x0), cons(x1, x2)))
   mark(zeros)
   mark(cons(x0, x1))
   mark(0)
   mark(U101(x0, x1, x2))
   mark(tt)
   mark(U102(x0, x1, x2))
   mark(isNatKind(x0))
   mark(U103(x0, x1, x2))
   mark(isNatIListKind(x0))
   mark(U104(x0, x1, x2))
   mark(U105(x0, x1))
   mark(isNat(x0))
   mark(U106(x0))
   mark(isNatIList(x0))
   mark(U11(x0, x1))
   mark(U12(x0, x1))
   mark(U111(x0, x1, x2))
   mark(U112(x0, x1, x2))
   mark(U113(x0, x1, x2))
   mark(U114(x0, x1))
   mark(s(x0))
   mark(length(x0))
   mark(U13(x0))
   mark(isNatList(x0))
   mark(U121(x0, x1))
   mark(U122(x0))
   mark(nil)
   mark(U131(x0, x1, x2, x3))
   mark(U132(x0, x1, x2, x3))
   mark(U133(x0, x1, x2, x3))
   mark(U134(x0, x1, x2, x3))
   mark(U135(x0, x1, x2, x3))
   mark(U136(x0, x1, x2, x3))
   mark(take(x0, x1))
   mark(U21(x0, x1))
   mark(U22(x0, x1))
   mark(U23(x0))
   mark(U31(x0, x1))
   mark(U32(x0, x1))
   mark(U33(x0))
   mark(U41(x0, x1, x2))
   mark(U42(x0, x1, x2))
   mark(U43(x0, x1, x2))
   mark(U44(x0, x1, x2))
   mark(U45(x0, x1))
   mark(U46(x0))
   mark(U51(x0, x1))
   mark(U52(x0))
   mark(U61(x0, x1))
   mark(U62(x0))
   mark(U71(x0))
   mark(U81(x0))
   mark(U91(x0, x1, x2))
   mark(U92(x0, x1, x2))
   mark(U93(x0, x1, x2))
   mark(U94(x0, x1, x2))
   mark(U95(x0, x1))
   mark(U96(x0))
   cons(mark(x0), x1)
   cons(x0, mark(x1))
   cons(active(x0), x1)
   cons(x0, active(x1))
   U101(mark(x0), x1, x2)
   U101(x0, mark(x1), x2)
   U101(x0, x1, mark(x2))
   U101(active(x0), x1, x2)
   U101(x0, active(x1), x2)
   U101(x0, x1, active(x2))
   U102(mark(x0), x1, x2)
   U102(x0, mark(x1), x2)
   U102(x0, x1, mark(x2))
   U102(active(x0), x1, x2)
   U102(x0, active(x1), x2)
   U102(x0, x1, active(x2))
   isNatKind(mark(x0))
   isNatKind(active(x0))
   U103(mark(x0), x1, x2)
   U103(x0, mark(x1), x2)
   U103(x0, x1, mark(x2))
   U103(active(x0), x1, x2)
   U103(x0, active(x1), x2)
   U103(x0, x1, active(x2))
   isNatIListKind(mark(x0))
   isNatIListKind(active(x0))
   U104(mark(x0), x1, x2)
   U104(x0, mark(x1), x2)
   U104(x0, x1, mark(x2))
   U104(active(x0), x1, x2)
   U104(x0, active(x1), x2)
   U104(x0, x1, active(x2))
   U105(mark(x0), x1)
   U105(x0, mark(x1))
   U105(active(x0), x1)
   U105(x0, active(x1))
   isNat(mark(x0))
   isNat(active(x0))
   U106(mark(x0))
   U106(active(x0))
   isNatIList(mark(x0))
   isNatIList(active(x0))
   U11(mark(x0), x1)
   U11(x0, mark(x1))
   U11(active(x0), x1)
   U11(x0, active(x1))
   U12(mark(x0), x1)
   U12(x0, mark(x1))
   U12(active(x0), x1)
   U12(x0, active(x1))
   U111(mark(x0), x1, x2)
   U111(x0, mark(x1), x2)
   U111(x0, x1, mark(x2))
   U111(active(x0), x1, x2)
   U111(x0, active(x1), x2)
   U111(x0, x1, active(x2))
   U112(mark(x0), x1, x2)
   U112(x0, mark(x1), x2)
   U112(x0, x1, mark(x2))
   U112(active(x0), x1, x2)
   U112(x0, active(x1), x2)
   U112(x0, x1, active(x2))
   U113(mark(x0), x1, x2)
   U113(x0, mark(x1), x2)
   U113(x0, x1, mark(x2))
   U113(active(x0), x1, x2)
   U113(x0, active(x1), x2)
   U113(x0, x1, active(x2))
   U114(mark(x0), x1)
   U114(x0, mark(x1))
   U114(active(x0), x1)
   U114(x0, active(x1))
   s(mark(x0))
   s(active(x0))
   length(mark(x0))
   length(active(x0))
   U13(mark(x0))
   U13(active(x0))
   isNatList(mark(x0))
   isNatList(active(x0))
   U121(mark(x0), x1)
   U121(x0, mark(x1))
   U121(active(x0), x1)
   U121(x0, active(x1))
   U122(mark(x0))
   U122(active(x0))
   U131(mark(x0), x1, x2, x3)
   U131(x0, mark(x1), x2, x3)
   U131(x0, x1, mark(x2), x3)
   U131(x0, x1, x2, mark(x3))
   U131(active(x0), x1, x2, x3)
   U131(x0, active(x1), x2, x3)
   U131(x0, x1, active(x2), x3)
   U131(x0, x1, x2, active(x3))
   U132(mark(x0), x1, x2, x3)
   U132(x0, mark(x1), x2, x3)
   U132(x0, x1, mark(x2), x3)
   U132(x0, x1, x2, mark(x3))
   U132(active(x0), x1, x2, x3)
   U132(x0, active(x1), x2, x3)
   U132(x0, x1, active(x2), x3)
   U132(x0, x1, x2, active(x3))
   U133(mark(x0), x1, x2, x3)
   U133(x0, mark(x1), x2, x3)
   U133(x0, x1, mark(x2), x3)
   U133(x0, x1, x2, mark(x3))
   U133(active(x0), x1, x2, x3)
   U133(x0, active(x1), x2, x3)
   U133(x0, x1, active(x2), x3)
   U133(x0, x1, x2, active(x3))
   U134(mark(x0), x1, x2, x3)
   U134(x0, mark(x1), x2, x3)
   U134(x0, x1, mark(x2), x3)
   U134(x0, x1, x2, mark(x3))
   U134(active(x0), x1, x2, x3)
   U134(x0, active(x1), x2, x3)
   U134(x0, x1, active(x2), x3)
   U134(x0, x1, x2, active(x3))
   U135(mark(x0), x1, x2, x3)
   U135(x0, mark(x1), x2, x3)
   U135(x0, x1, mark(x2), x3)
   U135(x0, x1, x2, mark(x3))
   U135(active(x0), x1, x2, x3)
   U135(x0, active(x1), x2, x3)
   U135(x0, x1, active(x2), x3)
   U135(x0, x1, x2, active(x3))
   U136(mark(x0), x1, x2, x3)
   U136(x0, mark(x1), x2, x3)
   U136(x0, x1, mark(x2), x3)
   U136(x0, x1, x2, mark(x3))
   U136(active(x0), x1, x2, x3)
   U136(x0, active(x1), x2, x3)
   U136(x0, x1, active(x2), x3)
   U136(x0, x1, x2, active(x3))
   take(mark(x0), x1)
   take(x0, mark(x1))
   take(active(x0), x1)
   take(x0, active(x1))
   U21(mark(x0), x1)
   U21(x0, mark(x1))
   U21(active(x0), x1)
   U21(x0, active(x1))
   U22(mark(x0), x1)
   U22(x0, mark(x1))
   U22(active(x0), x1)
   U22(x0, active(x1))
   U31(mark(x0), x1)
   U31(x0, mark(x1))
   U31(active(x0), x1)
   U31(x0, active(x1))
   U32(mark(x0), x1)
   U32(x0, mark(x1))
   U32(active(x0), x1)
   U32(x0, active(x1))
   U33(mark(x0))
   U33(active(x0))
   U41(mark(x0), x1, x2)
   U41(x0, mark(x1), x2)
   U41(x0, x1, mark(x2))
   U41(active(x0), x1, x2)
   U41(x0, active(x1), x2)
   U41(x0, x1, active(x2))
   U42(mark(x0), x1, x2)
   U42(x0, mark(x1), x2)
   U42(x0, x1, mark(x2))
   U42(active(x0), x1, x2)
   U42(x0, active(x1), x2)
   U42(x0, x1, active(x2))
   U43(mark(x0), x1, x2)
   U43(x0, mark(x1), x2)
   U43(x0, x1, mark(x2))
   U43(active(x0), x1, x2)
   U43(x0, active(x1), x2)
   U43(x0, x1, active(x2))
   U44(mark(x0), x1, x2)
   U44(x0, mark(x1), x2)
   U44(x0, x1, mark(x2))
   U44(active(x0), x1, x2)
   U44(x0, active(x1), x2)
   U44(x0, x1, active(x2))
   U45(mark(x0), x1)
   U45(x0, mark(x1))
   U45(active(x0), x1)
   U45(x0, active(x1))
   U46(mark(x0))
   U46(active(x0))
   U51(mark(x0), x1)
   U51(x0, mark(x1))
   U51(active(x0), x1)
   U51(x0, active(x1))
   U52(mark(x0))
   U52(active(x0))
   U61(mark(x0), x1)
   U61(x0, mark(x1))
   U61(active(x0), x1)
   U61(x0, active(x1))
   U62(mark(x0))
   U62(active(x0))
   U71(mark(x0))
   U71(active(x0))
   U81(mark(x0))
   U81(active(x0))
   U91(mark(x0), x1, x2)
   U91(x0, mark(x1), x2)
   U91(x0, x1, mark(x2))
   U91(active(x0), x1, x2)
   U91(x0, active(x1), x2)
   U91(x0, x1, active(x2))
   U92(mark(x0), x1, x2)
   U92(x0, mark(x1), x2)
   U92(x0, x1, mark(x2))
   U92(active(x0), x1, x2)
   U92(x0, active(x1), x2)
   U92(x0, x1, active(x2))
   U93(mark(x0), x1, x2)
   U93(x0, mark(x1), x2)
   U93(x0, x1, mark(x2))
   U93(active(x0), x1, x2)
   U93(x0, active(x1), x2)
   U93(x0, x1, active(x2))
   U94(mark(x0), x1, x2)
   U94(x0, mark(x1), x2)
   U94(x0, x1, mark(x2))
   U94(active(x0), x1, x2)
   U94(x0, active(x1), x2)
   U94(x0, x1, active(x2))
   U95(mark(x0), x1)
   U95(x0, mark(x1))
   U95(active(x0), x1)
   U95(x0, active(x1))
   U96(mark(x0))
   U96(active(x0))


----------------------------------------

(565)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(U23(X)) -> MARK(X)

R is empty.
The set Q consists of the following terms:

   U23(mark(x0))
   U23(active(x0))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(566) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*MARK(U23(X)) -> MARK(X)
The graph contains the following edges 1 > 1


----------------------------------------

(567)
YES