/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 712 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) UsableRulesProof [EQUIVALENT, 0 ms]
        (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) UsableRulesProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) UsableRulesProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (24) YES
    (25) QDP
        (26) UsableRulesProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (29) YES
    (30) QDP
        (31) UsableRulesProof [EQUIVALENT, 0 ms]
        (32) QDP
        (33) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (34) YES
    (35) QDP
        (36) UsableRulesProof [EQUIVALENT, 0 ms]
        (37) QDP
        (38) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (39) YES
    (40) QDP
        (41) UsableRulesProof [EQUIVALENT, 0 ms]
        (42) QDP
        (43) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (44) YES
    (45) QDP
        (46) UsableRulesProof [EQUIVALENT, 0 ms]
        (47) QDP
        (48) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (49) YES
    (50) QDP
        (51) UsableRulesProof [EQUIVALENT, 0 ms]
        (52) QDP
        (53) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (54) YES
    (55) QDP
        (56) UsableRulesProof [EQUIVALENT, 0 ms]
        (57) QDP
        (58) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (59) YES
    (60) QDP
        (61) UsableRulesProof [EQUIVALENT, 0 ms]
        (62) QDP
        (63) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (64) YES
    (65) QDP
        (66) UsableRulesProof [EQUIVALENT, 0 ms]
        (67) QDP
        (68) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (69) YES
    (70) QDP
        (71) UsableRulesProof [EQUIVALENT, 0 ms]
        (72) QDP
        (73) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (74) YES
    (75) QDP
        (76) UsableRulesProof [EQUIVALENT, 0 ms]
        (77) QDP
        (78) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (79) YES
    (80) QDP
        (81) UsableRulesProof [EQUIVALENT, 0 ms]
        (82) QDP
        (83) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (84) YES
    (85) QDP
        (86) UsableRulesProof [EQUIVALENT, 0 ms]
        (87) QDP
        (88) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (89) YES
    (90) QDP
        (91) UsableRulesProof [EQUIVALENT, 0 ms]
        (92) QDP
        (93) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (94) YES
    (95) QDP
        (96) UsableRulesProof [EQUIVALENT, 0 ms]
        (97) QDP
        (98) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (99) YES
    (100) QDP
        (101) UsableRulesProof [EQUIVALENT, 0 ms]
        (102) QDP
        (103) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (104) YES
    (105) QDP
        (106) UsableRulesProof [EQUIVALENT, 0 ms]
        (107) QDP
        (108) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (109) YES
    (110) QDP
        (111) UsableRulesProof [EQUIVALENT, 0 ms]
        (112) QDP
        (113) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (114) YES
    (115) QDP
        (116) UsableRulesProof [EQUIVALENT, 0 ms]
        (117) QDP
        (118) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (119) YES
    (120) QDP
        (121) UsableRulesProof [EQUIVALENT, 0 ms]
        (122) QDP
        (123) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (124) YES
    (125) QDP
        (126) UsableRulesProof [EQUIVALENT, 0 ms]
        (127) QDP
        (128) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (129) YES
    (130) QDP
        (131) UsableRulesProof [EQUIVALENT, 0 ms]
        (132) QDP
        (133) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (134) YES
    (135) QDP
        (136) UsableRulesProof [EQUIVALENT, 0 ms]
        (137) QDP
        (138) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (139) YES
    (140) QDP
        (141) UsableRulesProof [EQUIVALENT, 0 ms]
        (142) QDP
        (143) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (144) YES
    (145) QDP
        (146) UsableRulesProof [EQUIVALENT, 0 ms]
        (147) QDP
        (148) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (149) YES
    (150) QDP
        (151) UsableRulesProof [EQUIVALENT, 0 ms]
        (152) QDP
        (153) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (154) YES
    (155) QDP
        (156) UsableRulesProof [EQUIVALENT, 0 ms]
        (157) QDP
        (158) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (159) YES
    (160) QDP
        (161) UsableRulesProof [EQUIVALENT, 0 ms]
        (162) QDP
        (163) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (164) YES
    (165) QDP
        (166) UsableRulesProof [EQUIVALENT, 0 ms]
        (167) QDP
        (168) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (169) YES
    (170) QDP
        (171) UsableRulesProof [EQUIVALENT, 0 ms]
        (172) QDP
        (173) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (174) YES
    (175) QDP
        (176) UsableRulesProof [EQUIVALENT, 0 ms]
        (177) QDP
        (178) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (179) YES
    (180) QDP
        (181) UsableRulesProof [EQUIVALENT, 0 ms]
        (182) QDP
        (183) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (184) YES
    (185) QDP
        (186) UsableRulesProof [EQUIVALENT, 0 ms]
        (187) QDP
        (188) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (189) YES
    (190) QDP
        (191) UsableRulesProof [EQUIVALENT, 0 ms]
        (192) QDP
        (193) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (194) YES
    (195) QDP
        (196) UsableRulesProof [EQUIVALENT, 0 ms]
        (197) QDP
        (198) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (199) YES
    (200) QDP
        (201) UsableRulesProof [EQUIVALENT, 0 ms]
        (202) QDP
        (203) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (204) YES
    (205) QDP
        (206) UsableRulesProof [EQUIVALENT, 0 ms]
        (207) QDP
        (208) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (209) YES
    (210) QDP
        (211) UsableRulesProof [EQUIVALENT, 0 ms]
        (212) QDP
        (213) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (214) YES
    (215) QDP
        (216) UsableRulesProof [EQUIVALENT, 0 ms]
        (217) QDP
        (218) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (219) YES
    (220) QDP
        (221) UsableRulesProof [EQUIVALENT, 0 ms]
        (222) QDP
        (223) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (224) YES
    (225) QDP
        (226) UsableRulesProof [EQUIVALENT, 0 ms]
        (227) QDP
        (228) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (229) YES
    (230) QDP
        (231) UsableRulesProof [EQUIVALENT, 0 ms]
        (232) QDP
        (233) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (234) YES
    (235) QDP
        (236) UsableRulesProof [EQUIVALENT, 0 ms]
        (237) QDP
        (238) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (239) YES
    (240) QDP
        (241) UsableRulesProof [EQUIVALENT, 0 ms]
        (242) QDP
        (243) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (244) YES
    (245) QDP
        (246) UsableRulesProof [EQUIVALENT, 0 ms]
        (247) QDP
        (248) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (249) YES
    (250) QDP
        (251) UsableRulesProof [EQUIVALENT, 0 ms]
        (252) QDP
        (253) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (254) YES
    (255) QDP
        (256) UsableRulesProof [EQUIVALENT, 0 ms]
        (257) QDP
        (258) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (259) YES
    (260) QDP
        (261) UsableRulesProof [EQUIVALENT, 0 ms]
        (262) QDP
        (263) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (264) YES
    (265) QDP
        (266) UsableRulesProof [EQUIVALENT, 0 ms]
        (267) QDP
        (268) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (269) YES
    (270) QDP
        (271) UsableRulesProof [EQUIVALENT, 0 ms]
        (272) QDP
        (273) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (274) YES
    (275) QDP
        (276) QDPOrderProof [EQUIVALENT, 2277 ms]
        (277) QDP
        (278) QDPOrderProof [EQUIVALENT, 2885 ms]
        (279) QDP
        (280) QDPOrderProof [EQUIVALENT, 2911 ms]
        (281) QDP
        (282) QDPOrderProof [EQUIVALENT, 310 ms]
        (283) QDP
        (284) QDPOrderProof [EQUIVALENT, 335 ms]
        (285) QDP
        (286) QDPOrderProof [EQUIVALENT, 1893 ms]
        (287) QDP
        (288) QDPOrderProof [EQUIVALENT, 313 ms]
        (289) QDP
        (290) QDPOrderProof [EQUIVALENT, 264 ms]
        (291) QDP
        (292) QDPOrderProof [EQUIVALENT, 281 ms]
        (293) QDP
        (294) DependencyGraphProof [EQUIVALENT, 0 ms]
        (295) QDP
        (296) QDPOrderProof [EQUIVALENT, 229 ms]
        (297) QDP
        (298) QDPOrderProof [EQUIVALENT, 2075 ms]
        (299) QDP
        (300) QDPOrderProof [EQUIVALENT, 216 ms]
        (301) QDP
        (302) QDPOrderProof [EQUIVALENT, 2326 ms]
        (303) QDP
        (304) QDPOrderProof [EQUIVALENT, 198 ms]
        (305) QDP
        (306) QDPOrderProof [EQUIVALENT, 2104 ms]
        (307) QDP
        (308) QDPOrderProof [EQUIVALENT, 206 ms]
        (309) QDP
        (310) QDPOrderProof [EQUIVALENT, 1877 ms]
        (311) QDP
        (312) QDPOrderProof [EQUIVALENT, 210 ms]
        (313) QDP
        (314) QDPOrderProof [EQUIVALENT, 2399 ms]
        (315) QDP
        (316) QDPOrderProof [EQUIVALENT, 2295 ms]
        (317) QDP
        (318) QDPOrderProof [EQUIVALENT, 1746 ms]
        (319) QDP
        (320) QDPOrderProof [EQUIVALENT, 2282 ms]
        (321) QDP
        (322) QDPOrderProof [EQUIVALENT, 2591 ms]
        (323) QDP
        (324) QDPOrderProof [EQUIVALENT, 2697 ms]
        (325) QDP
        (326) QDPOrderProof [EQUIVALENT, 1755 ms]
        (327) QDP
        (328) QDPOrderProof [EQUIVALENT, 1897 ms]
        (329) QDP
        (330) QDPOrderProof [EQUIVALENT, 1989 ms]
        (331) QDP
        (332) QDPOrderProof [EQUIVALENT, 1799 ms]
        (333) QDP
        (334) QDPOrderProof [EQUIVALENT, 190 ms]
        (335) QDP
        (336) QDPOrderProof [EQUIVALENT, 1863 ms]
        (337) QDP
        (338) QDPOrderProof [EQUIVALENT, 2186 ms]
        (339) QDP
        (340) QDPOrderProof [EQUIVALENT, 192 ms]
        (341) QDP
        (342) QDPOrderProof [EQUIVALENT, 2283 ms]
        (343) QDP
        (344) QDPOrderProof [EQUIVALENT, 1669 ms]
        (345) QDP
        (346) QDPOrderProof [EQUIVALENT, 2029 ms]
        (347) QDP
        (348) QDPOrderProof [EQUIVALENT, 1494 ms]
        (349) QDP
        (350) QDPOrderProof [EQUIVALENT, 2116 ms]
        (351) QDP
        (352) QDPOrderProof [EQUIVALENT, 193 ms]
        (353) QDP
        (354) QDPOrderProof [EQUIVALENT, 193 ms]
        (355) QDP
        (356) QDPOrderProof [EQUIVALENT, 1942 ms]
        (357) QDP
        (358) QDPOrderProof [EQUIVALENT, 1677 ms]
        (359) QDP
        (360) QDPOrderProof [EQUIVALENT, 195 ms]
        (361) QDP
        (362) QDPOrderProof [EQUIVALENT, 1792 ms]
        (363) QDP
        (364) QDPOrderProof [EQUIVALENT, 1948 ms]
        (365) QDP
        (366) QDPOrderProof [EQUIVALENT, 131 ms]
        (367) QDP
        (368) QDPOrderProof [EQUIVALENT, 168 ms]
        (369) QDP
        (370) QDPOrderProof [EQUIVALENT, 1699 ms]
        (371) QDP
        (372) QDPOrderProof [EQUIVALENT, 198 ms]
        (373) QDP
        (374) QDPOrderProof [EQUIVALENT, 188 ms]
        (375) QDP
        (376) QDPOrderProof [EQUIVALENT, 1590 ms]
        (377) QDP
        (378) QDPOrderProof [EQUIVALENT, 2168 ms]
        (379) QDP
        (380) QDPOrderProof [EQUIVALENT, 1892 ms]
        (381) QDP
        (382) QDPOrderProof [EQUIVALENT, 1694 ms]
        (383) QDP
        (384) QDPOrderProof [EQUIVALENT, 2000 ms]
        (385) QDP
        (386) QDPOrderProof [EQUIVALENT, 1618 ms]
        (387) QDP
        (388) QDPOrderProof [EQUIVALENT, 2758 ms]
        (389) QDP
        (390) QDPOrderProof [EQUIVALENT, 1545 ms]
        (391) QDP
        (392) QDPOrderProof [EQUIVALENT, 1435 ms]
        (393) QDP
        (394) QDPOrderProof [EQUIVALENT, 2012 ms]
        (395) QDP
        (396) QDPOrderProof [EQUIVALENT, 1434 ms]
        (397) QDP
        (398) QDPOrderProof [EQUIVALENT, 193 ms]
        (399) QDP
        (400) QDPOrderProof [EQUIVALENT, 1493 ms]
        (401) QDP
        (402) QDPOrderProof [EQUIVALENT, 1352 ms]
        (403) QDP
        (404) QDPOrderProof [EQUIVALENT, 1728 ms]
        (405) QDP
        (406) QDPOrderProof [EQUIVALENT, 1370 ms]
        (407) QDP
        (408) QDPOrderProof [EQUIVALENT, 1325 ms]
        (409) QDP
        (410) QDPOrderProof [EQUIVALENT, 2494 ms]
        (411) QDP
        (412) QDPOrderProof [EQUIVALENT, 1360 ms]
        (413) QDP
        (414) QDPOrderProof [EQUIVALENT, 1482 ms]
        (415) QDP
        (416) QDPOrderProof [EQUIVALENT, 3156 ms]
        (417) QDP
        (418) QDPOrderProof [EQUIVALENT, 1406 ms]
        (419) QDP
        (420) QDPOrderProof [EQUIVALENT, 1493 ms]
        (421) QDP
        (422) QDPOrderProof [EQUIVALENT, 1355 ms]
        (423) QDP
        (424) QDPOrderProof [EQUIVALENT, 1271 ms]
        (425) QDP
        (426) QDPOrderProof [EQUIVALENT, 1232 ms]
        (427) QDP
        (428) QDPOrderProof [EQUIVALENT, 384 ms]
        (429) QDP
        (430) PisEmptyProof [EQUIVALENT, 0 ms]
        (431) 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)

Q is empty.

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

(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)

Q is empty.
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)

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

(6) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, 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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(8) 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


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

(9)
YES

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

(10)
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)

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

(11) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(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)

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

(13) 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


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

(14)
YES

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

(15)
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)

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

(16) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(17)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(18) 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


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(19)
YES

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(20)
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)

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

(21) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(22)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(23) 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


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

(24)
YES

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

(25)
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)

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

(26) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(27)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(28) 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


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

(29)
YES

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

(30)
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)

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

(31) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(32)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(33) 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


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

(34)
YES

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

(35)
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)

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

(36) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(37)
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.
Q is empty.
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:
*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


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

(39)
YES

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

(40)
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)

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

(41) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(42)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(43) 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


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

(44)
YES

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

(45)
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)

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

(46) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(47)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(48) 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


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

(49)
YES

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

(50)
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)

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

(51) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(52)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(53) 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


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

(54)
YES

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

(55)
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)

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

(56) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(57)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(58) 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


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

(59)
YES

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

(60)
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)

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

(61) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(62)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(63) 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


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

(64)
YES

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

(65)
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)

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

(66) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(67)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(68) 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


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

(69)
YES

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

(70)
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)

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

(71) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(72)
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.
Q is empty.
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:
*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


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

(74)
YES

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

(75)
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)

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

(76) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(77)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(78) 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


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

(79)
YES

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

(80)
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)

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

(81) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(82)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(83) 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


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

(84)
YES

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

(85)
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)

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

(86) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(87)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(88) 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


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

(89)
YES

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

(90)
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)

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

(91) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(92)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(93) 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


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

(94)
YES

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

(95)
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)

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

(96) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(97)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(98) 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


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

(99)
YES

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

(100)
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)

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

(101) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(102)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(103) 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


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

(104)
YES

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

(105)
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)

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

(106) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(107)
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.
Q is empty.
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:
*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


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

(109)
YES

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

(110)
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)

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

(111) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(112)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(113) 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


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

(114)
YES

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

(115)
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)

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

(116) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(117)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(118) 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


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

(119)
YES

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

(120)
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)

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

(121) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(122)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(123) 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


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

(124)
YES

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

(125)
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)

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

(126) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(127)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(128) 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


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

(129)
YES

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

(130)
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)

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

(131) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(132)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(133) 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


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

(134)
YES

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

(135)
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)

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

(136) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(137)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(138) 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


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

(139)
YES

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

(140)
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)

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

(141) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(142)
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.
Q is empty.
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:
*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


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

(144)
YES

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

(145)
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)

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

(146) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(147)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(148) 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


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

(149)
YES

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

(150)
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)

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

(151) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(152)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(153) 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


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

(154)
YES

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

(155)
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)

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

(156) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(157)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(158) 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


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

(159)
YES

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

(160)
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)

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

(161) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(162)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(163) 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


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

(164)
YES

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

(165)
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)

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

(166) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(167)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(168) 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


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

(169)
YES

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

(170)
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)

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

(171) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(172)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(173) 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


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

(174)
YES

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

(175)
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)

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

(176) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(177)
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.
Q is empty.
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:
*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


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

(179)
YES

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

(180)
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)

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

(181) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(182)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(183) 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


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

(184)
YES

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

(185)
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)

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

(186) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(187)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(188) 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


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

(189)
YES

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

(190)
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)

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

(191) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(192)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(193) 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


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

(194)
YES

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

(195)
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)

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

(196) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(197)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(198) 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


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

(199)
YES

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

(200)
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)

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

(201) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(202)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(203) 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


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

(204)
YES

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

(205)
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)

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

(206) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(207)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(208) 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


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

(209)
YES

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

(210)
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)

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

(211) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(212)
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.
Q is empty.
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:
*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


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

(214)
YES

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

(215)
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)

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

(216) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(217)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(218) 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


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

(219)
YES

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

(220)
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)

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

(221) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(222)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(223) 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


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

(224)
YES

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

(225)
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)

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

(226) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(227)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(228) 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


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

(229)
YES

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

(230)
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)

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

(231) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(232)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(233) 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


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

(234)
YES

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

(235)
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)

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

(236) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(237)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(238) 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


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

(239)
YES

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

(240)
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)

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

(241) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(242)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(243) 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


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

(244)
YES

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

(245)
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)

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

(246) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(247)
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.
Q is empty.
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:
*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


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

(249)
YES

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

(250)
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)

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

(251) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(252)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(253) 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


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

(254)
YES

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

(255)
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)

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

(256) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(257)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(258) 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


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

(259)
YES

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

(260)
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)

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

(261) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(262)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(263) 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


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

(264)
YES

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

(265)
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)

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

(266) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(267)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(268) 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


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

(269)
YES

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

(270)
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)

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

(271) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(272)
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.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(273) 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


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

(274)
YES

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

(275)
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(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(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U106(X)) -> ACTIVE(U106(mark(X)))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U121(tt, IL)) -> MARK(U122(isNatIListKind(IL)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   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(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U114(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(length(X)) -> MARK(X)
   MARK(U13(X)) -> ACTIVE(U13(mark(X)))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U121(X1, X2)) -> MARK(X1)
   MARK(U122(X)) -> ACTIVE(U122(mark(X)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U122(X)) -> MARK(X)
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U131(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U132(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U133(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U134(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U135(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U136(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> ACTIVE(U33(mark(X)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> ACTIVE(U46(mark(X)))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   MARK(U52(X)) -> MARK(X)
   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(U62(X)) -> ACTIVE(U62(mark(X)))
   ACTIVE(take(0, IL)) -> MARK(U121(isNatIList(IL), IL))
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> ACTIVE(U71(mark(X)))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> ACTIVE(U81(mark(X)))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(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)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(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)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> ACTIVE(U96(mark(X)))
   MARK(U96(X)) -> MARK(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)

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

(276) 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(s(X)) -> ACTIVE(s(mark(X)))
   MARK(U13(X)) -> ACTIVE(U13(mark(X)))
   MARK(U122(X)) -> ACTIVE(U122(mark(X)))
   MARK(U23(X)) -> ACTIVE(U23(mark(X)))
   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) ) = 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) ) = 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( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = max{0, -2}
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) ) = 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( U52_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 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( U96_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( take_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = max{0, -2}
POL( zeros ) = 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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   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)


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

(277)
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(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(U102(tt, V1, V2)) -> MARK(U103(isNatIListKind(V2), V1, V2))
   MARK(U101(X1, X2, X3)) -> MARK(X1)
   MARK(U102(X1, X2, X3)) -> ACTIVE(U102(mark(X1), X2, X3))
   ACTIVE(U103(tt, V1, V2)) -> MARK(U104(isNatIListKind(V2), V1, V2))
   MARK(U102(X1, X2, X3)) -> MARK(X1)
   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U103(X1, X2, X3)) -> ACTIVE(U103(mark(X1), X2, X3))
   ACTIVE(U105(tt, V2)) -> MARK(U106(isNatIList(V2)))
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   MARK(U104(X1, X2, X3)) -> ACTIVE(U104(mark(X1), X2, X3))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(U104(X1, X2, X3)) -> MARK(X1)
   MARK(U105(X1, X2)) -> ACTIVE(U105(mark(X1), X2))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(U105(X1, X2)) -> MARK(X1)
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U106(X)) -> MARK(X)
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U121(tt, IL)) -> MARK(U122(isNatIListKind(IL)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U131(tt, IL, M, N)) -> MARK(U132(isNatIListKind(IL), IL, M, N))
   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(U111(X1, X2, X3)) -> MARK(X1)
   MARK(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   ACTIVE(U133(tt, IL, M, N)) -> MARK(U134(isNatKind(M), IL, M, N))
   MARK(U112(X1, X2, X3)) -> MARK(X1)
   MARK(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), IL, M, N))
   MARK(U113(X1, X2, X3)) -> MARK(X1)
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U114(X1, X2)) -> MARK(X1)
   ACTIVE(U136(tt, IL, M, N)) -> MARK(cons(N, take(M, IL)))
   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   MARK(length(X)) -> MARK(X)
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   MARK(U121(X1, X2)) -> ACTIVE(U121(mark(X1), X2))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U121(X1, X2)) -> MARK(X1)
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   MARK(U122(X)) -> MARK(X)
   MARK(U131(X1, X2, X3, X4)) -> ACTIVE(U131(mark(X1), X2, X3, X4))
   ACTIVE(U42(tt, V1, V2)) -> MARK(U43(isNatIListKind(V2), V1, V2))
   MARK(U131(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U132(X1, X2, X3, X4)) -> ACTIVE(U132(mark(X1), X2, X3, X4))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U132(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U133(X1, X2, X3, X4)) -> ACTIVE(U133(mark(X1), X2, X3, X4))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U133(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U134(X1, X2, X3, X4)) -> ACTIVE(U134(mark(X1), X2, X3, X4))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U134(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U135(X1, X2, X3, X4)) -> MARK(X1)
   MARK(U136(X1, X2, X3, X4)) -> ACTIVE(U136(mark(X1), X2, X3, X4))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U136(X1, X2, X3, X4)) -> MARK(X1)
   MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2)))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(take(X1, X2)) -> MARK(X1)
   MARK(take(X1, X2)) -> MARK(X2)
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U22(X1, X2)) -> MARK(X1)
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> ACTIVE(U33(mark(X)))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U45(X1, X2)) -> MARK(X1)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   MARK(U52(X)) -> MARK(X)
   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(take(0, IL)) -> MARK(U121(isNatIList(IL), IL))
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> ACTIVE(U71(mark(X)))
   ACTIVE(take(s(M), cons(N, IL))) -> MARK(U131(isNatIList(IL), IL, M, N))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> ACTIVE(U81(mark(X)))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(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)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(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)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> ACTIVE(U96(mark(X)))
   MARK(U96(X)) -> MARK(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)

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

(278) 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(U33(X)) -> ACTIVE(U33(mark(X)))
   MARK(U52(X)) -> ACTIVE(U52(mark(X)))
   MARK(U71(X)) -> ACTIVE(U71(mark(X)))
   MARK(U96(X)) -> ACTIVE(U96(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( 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( U51_2(x_1, x_2) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U71_1(x_1) ) = max{0, -2}
POL( U81_1(x_1) ) = 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( U96_1(x_1) ) = 1
POL( cons_2(x_1, x_2) ) = 0
POL( length_1(x_1) ) = 2
POL( take_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( active_1(x_1) ) = max{0, -2}
POL( zeros ) = 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) ) = 2
POL( isNatIList_1(x_1) ) = 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) ) = 2
POL( U122_1(x_1) ) = max{0, x_1 - 2}
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( U46_1(x_1) ) = 2
POL( U62_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:

   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(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)
   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)
   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)
   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)
   U96(active(X)) -> U96(X)
   U96(mark(X)) -> U96(X)
   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)


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

(279)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

(280) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   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) ) = max{0, x_1 - 1}
POL( MARK_1(x_1) ) = 1
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U134_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( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U81_1(x_1) ) = 0
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 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( 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( U135_4(x_1, ..., x_4) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_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( take_2(x_1, x_2) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 2
POL( 0 ) = 0
POL( tt ) = 0
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) ) = 2
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U122_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = 2
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) ) = 2
POL( U71_1(x_1) ) = x_1 + 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(281)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

(282) 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)) -> 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)
   ACTIVE(take(0, IL)) -> MARK(U121(isNatIList(IL), 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)) = 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:

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


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

(283)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

(284) 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:

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


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

(285)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

(286) 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) ) = max{0, 2x_1 - 2}
POL( MARK_1(x_1) ) = 2
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 2
POL( U134_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( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U81_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( active_1(x_1) ) = 1
POL( mark_1(x_1) ) = 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( 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( U135_4(x_1, ..., x_4) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_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( take_2(x_1, x_2) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( zeros ) = 2
POL( 0 ) = 0
POL( tt ) = 0
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, 2x_1 - 2}
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 2
POL( U122_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, 2x_1 - 1}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = x_1 + 1
POL( U96_1(x_1) ) = 2
POL( U71_1(x_1) ) = 0
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(287)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

(288) 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)) = 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
   POL(U122(x_1)) = 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:

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


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

(289)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

(290) 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)) = 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
   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:

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


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

(291)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

(292) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(zeros) -> MARK(cons(0, zeros))
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)) = 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)) = 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)) = x_2
   POL(tt) = 0
   POL(zeros) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


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

(293)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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

(294) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(295)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(296) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

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


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

(297)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(298) 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) ) = 0
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( 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) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 1}
POL( active_1(x_1) ) = 2
POL( zeros ) = 0
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) ) = 0
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = 1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = 2x_1 + 2
POL( U122_1(x_1) ) = x_1
POL( U23_1(x_1) ) = 1
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2
POL( U52_1(x_1) ) = 1
POL( U62_1(x_1) ) = 1
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( U81_1(x_1) ) = 0
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:

   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)
   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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(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)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(299)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

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


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

(301)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(302) 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( 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( 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) ) = 1
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( active_1(x_1) ) = max{0, -2}
POL( zeros ) = 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 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) ) = max{0, -2}
POL( U13_1(x_1) ) = x_1 + 2
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U23_1(x_1) ) = 2
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U46_1(x_1) ) = 1
POL( U52_1(x_1) ) = 2x_1 + 2
POL( U62_1(x_1) ) = max{0, -2}
POL( U96_1(x_1) ) = 2
POL( U71_1(x_1) ) = x_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:

   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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(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)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(303)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(304) 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)) -> MARK(X1)
   ACTIVE(U104(tt, V1, V2)) -> MARK(U105(isNat(V1), V2))
   MARK(U103(X1, X2, X3)) -> MARK(X1)
   MARK(U104(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_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)) = 1 + x_1 + x_2 + x_3
   POL(U104(x_1, x_2, x_3)) = 1 + 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)) = 0
   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(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(zeros) -> active(zeros)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(isNat(X)) -> active(isNat(X))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   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(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(s(X)) -> active(s(mark(X)))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   mark(length(X)) -> active(length(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(isNatList(X)) -> active(isNatList(X))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U33(X)) -> active(U33(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U52(X)) -> active(U52(mark(X)))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U62(X)) -> active(U62(mark(X)))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U71(X)) -> active(U71(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   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)))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   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)
   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)
   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)
   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)
   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)
   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)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(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)


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

(305)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(306) 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))
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( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 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) ) = 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( mark_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( active_1(x_1) ) = 2x_1 + 2
POL( zeros ) = 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 - 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) ) = 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_2 - 2}
POL( U122_1(x_1) ) = x_1 + 2
POL( take_2(x_1, x_2) ) = x_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, -2}
POL( U62_1(x_1) ) = max{0, -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) ) = 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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(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)
   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)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(307)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(308) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

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


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

(309)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(310) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   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) ) = max{0, x_1 - 1}
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_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) ) = max{0, -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( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( tt ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 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, 2x_1 - 2}
POL( U13_1(x_1) ) = x_1 + 2
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = x_1 + 2
POL( U122_1(x_1) ) = 2
POL( take_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U23_1(x_1) ) = max{0, -2}
POL( U33_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2
POL( U52_1(x_1) ) = 1
POL( U62_1(x_1) ) = 1
POL( U96_1(x_1) ) = 2
POL( U71_1(x_1) ) = max{0, -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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(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)
   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)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(311)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(312) 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)) = 0
   POL(U104(x_1, x_2, x_3)) = 0
   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(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(zeros) -> active(zeros)
   mark(U101(X1, X2, X3)) -> active(U101(mark(X1), X2, X3))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   mark(U102(X1, X2, X3)) -> active(U102(mark(X1), X2, X3))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   mark(isNatKind(X)) -> active(isNatKind(X))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   mark(U103(X1, X2, X3)) -> active(U103(mark(X1), X2, X3))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   mark(isNatIListKind(X)) -> active(isNatIListKind(X))
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   mark(U104(X1, X2, X3)) -> active(U104(mark(X1), X2, X3))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   mark(U105(X1, X2)) -> active(U105(mark(X1), X2))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   mark(isNat(X)) -> active(isNat(X))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   mark(U106(X)) -> active(U106(mark(X)))
   active(U114(tt, L)) -> mark(s(length(L)))
   mark(isNatIList(X)) -> active(isNatIList(X))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   mark(U12(X1, X2)) -> active(U12(mark(X1), X2))
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   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(U112(X1, X2, X3)) -> active(U112(mark(X1), X2, X3))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   mark(U113(X1, X2, X3)) -> active(U113(mark(X1), X2, X3))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   mark(U114(X1, X2)) -> active(U114(mark(X1), X2))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   mark(s(X)) -> active(s(mark(X)))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   mark(length(X)) -> active(length(mark(X)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   mark(U13(X)) -> active(U13(mark(X)))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   mark(isNatList(X)) -> active(isNatList(X))
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   mark(U121(X1, X2)) -> active(U121(mark(X1), X2))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   mark(U122(X)) -> active(U122(mark(X)))
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   mark(U131(X1, X2, X3, X4)) -> active(U131(mark(X1), X2, X3, X4))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   mark(U132(X1, X2, X3, X4)) -> active(U132(mark(X1), X2, X3, X4))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   mark(U133(X1, X2, X3, X4)) -> active(U133(mark(X1), X2, X3, X4))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   mark(U134(X1, X2, X3, X4)) -> active(U134(mark(X1), X2, X3, X4))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   mark(U135(X1, X2, X3, X4)) -> active(U135(mark(X1), X2, X3, X4))
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   mark(U136(X1, X2, X3, X4)) -> active(U136(mark(X1), X2, X3, X4))
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   mark(take(X1, X2)) -> active(take(mark(X1), mark(X2)))
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   mark(U21(X1, X2)) -> active(U21(mark(X1), X2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   mark(U22(X1, X2)) -> active(U22(mark(X1), X2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   mark(U23(X)) -> active(U23(mark(X)))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   mark(U31(X1, X2)) -> active(U31(mark(X1), X2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   mark(U32(X1, X2)) -> active(U32(mark(X1), X2))
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   mark(U33(X)) -> active(U33(mark(X)))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   mark(U42(X1, X2, X3)) -> active(U42(mark(X1), X2, X3))
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
   mark(U43(X1, X2, X3)) -> active(U43(mark(X1), X2, X3))
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   mark(U44(X1, X2, X3)) -> active(U44(mark(X1), X2, X3))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   mark(U45(X1, X2)) -> active(U45(mark(X1), X2))
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   mark(U46(X)) -> active(U46(mark(X)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   mark(U51(X1, X2)) -> active(U51(mark(X1), X2))
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   mark(U52(X)) -> active(U52(mark(X)))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   mark(U61(X1, X2)) -> active(U61(mark(X1), X2))
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   mark(U62(X)) -> active(U62(mark(X)))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   mark(U71(X)) -> active(U71(mark(X)))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   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)))
   mark(0) -> active(0)
   mark(tt) -> active(tt)
   mark(nil) -> active(nil)
   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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   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)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   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)
   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)
   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)
   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)
   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)
   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)
   U122(active(X)) -> U122(X)
   U122(mark(X)) -> U122(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)


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

(313)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(314) 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( U102_3(x_1, ..., x_3) ) = 2
POL( U103_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( 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) ) = 2
POL( cons_2(x_1, x_2) ) = x_1 + x_2
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( tt ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 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, -2}
POL( U13_1(x_1) ) = max{0, x_1 - 2}
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 2x_1 + 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( take_2(x_1, x_2) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, 2x_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) ) = max{0, x_1 - 2}
POL( U96_1(x_1) ) = max{0, 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) ) = 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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(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)
   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)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(315)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(316) 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(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, -2}
POL( U102_3(x_1, ..., x_3) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = 0
POL( U105_2(x_1, x_2) ) = 2x_1 + 2x_2 + 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) ) = 0
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 + 1
POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2x_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( U135_4(x_1, ..., x_4) ) = max{0, -2}
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) ) = 2x_1 + 1
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = x_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) ) = 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) ) = max{0, -2}
POL( mark_1(x_1) ) = 2x_1
POL( cons_2(x_1, x_2) ) = 2x_1 + 2
POL( active_1(x_1) ) = 2x_1 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, -2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, -2}
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) ) = max{0, x_1 + x_2 - 2}
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U132_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) ) = max{0, x_1 - 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:

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(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)
   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)
   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)
   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)
   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)
   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)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   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)
   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)
   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)
   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)
   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)


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

(317)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(318) 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 Order [NEGPOLO,POLO] with Interpretation:

POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, -2}
POL( U105_2(x_1, x_2) ) = 0
POL( U11_2(x_1, x_2) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 0
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 0
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = max{0, -2}
POL( U135_4(x_1, ..., x_4) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = max{0, -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) ) = 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) ) = 0
POL( U61_2(x_1, x_2) ) = 0
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) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( tt ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( isNat_1(x_1) ) = 2
POL( U106_1(x_1) ) = 0
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) ) = 0
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) ) = 2x_1
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + 2x_3 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = x_2 + 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, 2x_1 - 2}
POL( U62_1(x_1) ) = max{0, x_1 - 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 ) = 2
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:

   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)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(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)
   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)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(319)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(320) 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))
   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( U102_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 2
POL( U103_3(x_1, ..., x_3) ) = max{0, -2}
POL( U105_2(x_1, x_2) ) = 2x_1 + 2x_2 + 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) ) = 2x_2 + x_3 + x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = 0
POL( U135_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) ) = 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) ) = 2x_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) ) = 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) ) = 0
POL( mark_1(x_1) ) = 2x_1 + 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( active_1(x_1) ) = x_1 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, -2}
POL( tt ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( isNatIListKind_1(x_1) ) = 0
POL( U104_3(x_1, ..., x_3) ) = max{0, -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) ) = x_1 + 1
POL( U13_1(x_1) ) = x_1 + 1
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U122_1(x_1) ) = max{0, -2}
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + x_3 + 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, -2}
POL( take_2(x_1, x_2) ) = 2x_2 + 2
POL( U23_1(x_1) ) = 2x_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) ) = x_1 + 1
POL( U71_1(x_1) ) = x_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:

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(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)
   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)
   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)
   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)
   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)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   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)
   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)
   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)
   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)
   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)


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

(321)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(322) 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))
   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( MARK_1(x_1) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = max{0, -2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U11_2(x_1, x_2) ) = 1
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = 0
POL( U32_2(x_1, x_2) ) = 0
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = 0
POL( U94_3(x_1, ..., x_3) ) = 1
POL( isNatIListKind_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U101_3(x_1, ..., x_3) ) = 2
POL( U114_2(x_1, x_2) ) = 0
POL( U134_4(x_1, ..., x_4) ) = 1
POL( U136_4(x_1, ..., x_4) ) = 2
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 1
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 0
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = max{0, -2}
POL( U81_1(x_1) ) = max{0, x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 1
POL( isNatKind_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = 1
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U135_4(x_1, ..., x_4) ) = 0
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U105_2(x_1, x_2) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 0
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) ) = max{0, x_1 + x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2x_4 + 2
POL( U132_4(x_1, ..., x_4) ) = 2x_3 + 2
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2x_3 + 2x_4 + 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, 2x_1 - 2}
POL( U96_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(323)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(324) 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(U106(X)) -> MARK(X)
   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, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U101_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 1
POL( U136_4(x_1, ..., x_4) ) = 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_1 + 1
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1
POL( mark_1(x_1) ) = 2x_1
POL( U104_3(x_1, ..., x_3) ) = max{0, -2}
POL( U11_2(x_1, x_2) ) = 2x_1 + 1
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U12_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( U43_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U52_1(x_1) ) = 2x_1 + 1
POL( U62_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatIListKind_1(x_1) ) = 0
POL( U103_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U111_3(x_1, ..., x_3) ) = 1
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 2x_1 + 1
POL( U95_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U105_2(x_1, x_2) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 0
POL( U106_1(x_1) ) = x_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 + 2x_2 + 2
POL( U122_1(x_1) ) = x_1
POL( U131_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U33_1(x_1) ) = x_1 + 1
POL( U46_1(x_1) ) = 2x_1 + 1
POL( U96_1(x_1) ) = x_1 + 1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(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)
   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)
   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)
   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)
   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)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   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)
   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)
   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)
   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)
   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)


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

(325)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(326) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U134(tt, IL, M, N)) -> MARK(U135(isNat(N), 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( MARK_1(x_1) ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U136_4(x_1, ..., x_4) ) = x_4 + 2
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = 0
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U81_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 0
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = 2x_1 + 2
POL( U62_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 2
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 0
POL( U113_3(x_1, ..., x_3) ) = 0
POL( U135_4(x_1, ..., x_4) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U106_1(x_1) ) = max{0, 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, 2x_1 + x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + x_3 + 2x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = x_1 + x_2 + 2x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = x_1 + 2
POL( take_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U96_1(x_1) ) = 1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(327)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

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)

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

(328) 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( MARK_1(x_1) ) = max{0, -2}
POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U136_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U81_1(x_1) ) = 0
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U11_2(x_1, x_2) ) = 2
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = 2
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U62_1(x_1) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = max{0, x_1 - 1}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U135_4(x_1, ..., x_4) ) = 0
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = 2
POL( zeros ) = 0
POL( 0 ) = 2
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U103_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 1
POL( U105_2(x_1, x_2) ) = x_2 + 2
POL( U106_1(x_1) ) = max{0, 2x_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) ) = 0
POL( U121_2(x_1, x_2) ) = 2x_1 + 2
POL( U122_1(x_1) ) = 0
POL( U131_4(x_1, ..., x_4) ) = x_1 + x_4
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 + x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = x_2 + 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = max{0, x_1 - 2}
POL( U96_1(x_1) ) = 0
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(329)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U12(X1, X2)) -> MARK(X1)
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   MARK(U135(X1, X2, X3, X4)) -> ACTIVE(U135(mark(X1), X2, X3, X4))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(330) 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, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U101_3(x_1, ..., x_3) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U136_4(x_1, ..., x_4) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = x_1 + 1
POL( U22_2(x_1, x_2) ) = x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = x_1 + 1
POL( U42_3(x_1, ..., x_3) ) = x_1 + 1
POL( U51_2(x_1, x_2) ) = x_1 + 1
POL( U61_2(x_1, x_2) ) = x_1 + 1
POL( U81_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = x_1 + 2
POL( U11_2(x_1, x_2) ) = x_1 + 1
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = 2x_1 + 1
POL( U31_2(x_1, x_2) ) = x_1 + 1
POL( U32_2(x_1, x_2) ) = x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = x_1 + 1
POL( U52_1(x_1) ) = x_1 + 1
POL( U62_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatIListKind_1(x_1) ) = 0
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U45_2(x_1, x_2) ) = 2x_1 + 1
POL( U95_2(x_1, x_2) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U135_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + x_3 + 2x_4 + 2
POL( length_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 1
POL( 0 ) = 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = x_2 + 2
POL( U103_3(x_1, ..., x_3) ) = 2x_1 + 2
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( U106_1(x_1) ) = 2
POL( s_1(x_1) ) = x_1 + 1
POL( U13_1(x_1) ) = x_1 + 1
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, x_2 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + 2x_3 + 2x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = x_1 + x_2 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 + x_4 - 2}
POL( take_2(x_1, x_2) ) = 2x_1 + 2
POL( U33_1(x_1) ) = 2x_1 + 1
POL( U96_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   U13(active(X)) -> U13(X)
   U13(mark(X)) -> U13(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)
   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)
   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)
   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)
   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)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   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)
   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)
   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)
   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)
   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)


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

(331)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U12(X1, X2)) -> MARK(X1)
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), IL, M, N))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(332) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U135(tt, IL, M, N)) -> MARK(U136(isNatKind(N), 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( MARK_1(x_1) ) = max{0, -2}
POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( U21_2(x_1, x_2) ) = 1
POL( U22_2(x_1, x_2) ) = 1
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 1
POL( U61_2(x_1, x_2) ) = max{0, -2}
POL( U81_1(x_1) ) = max{0, 2x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = 1
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( mark_1(x_1) ) = 2
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U12_2(x_1, x_2) ) = 0
POL( U31_2(x_1, x_2) ) = 1
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U52_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 0
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U23_1(x_1) ) = x_1 + 2
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = 1
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 1
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U106_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) ) = 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 + 2x_3 + x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = x_1 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U33_1(x_1) ) = max{0, 2x_1 - 2}
POL( U96_1(x_1) ) = max{0, -2}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(333)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U11(X1, X2)) -> MARK(X1)
   MARK(U12(X1, X2)) -> ACTIVE(U12(mark(X1), X2))
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U12(X1, X2)) -> MARK(X1)
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(334) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(U11(X1, X2)) -> MARK(X1)
   ACTIVE(U12(tt, V1)) -> MARK(U13(isNatList(V1)))
   MARK(U12(X1, X2)) -> MARK(X1)
   ACTIVE(isNatList(take(V1, V2))) -> MARK(U101(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)) = 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)) = 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)) = 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 + 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)) = 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:

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


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

(335)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(U13(X)) -> MARK(X)
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(336) 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( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U114_2(x_1, x_2) ) = 1
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_1 + 1
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = 2x_1
POL( U11_2(x_1, x_2) ) = 1
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U12_2(x_1, x_2) ) = 1
POL( U31_2(x_1, x_2) ) = 2x_1 + 1
POL( U32_2(x_1, x_2) ) = 2x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U52_1(x_1) ) = x_1 + 1
POL( U62_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatIListKind_1(x_1) ) = 0
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U45_2(x_1, x_2) ) = 2x_1 + 1
POL( U95_2(x_1, x_2) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = 0
POL( length_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U106_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = 2x_1 + 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) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2x_1 + 2x_3 + 2x_4 + 2
POL( U132_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 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) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( U136_4(x_1, ..., x_4) ) = 2x_3 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = x_1 + 1
POL( U96_1(x_1) ) = x_1 + 1
POL( nil ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   U52(active(X)) -> U52(X)
   U52(mark(X)) -> U52(X)
   U62(active(X)) -> U62(X)
   U62(mark(X)) -> U62(X)
   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)
   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)
   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)
   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)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(337)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(338) 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, x_1 - 1}
POL( MARK_1(x_1) ) = 1
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( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 2
POL( U81_1(x_1) ) = 0
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( U45_2(x_1, x_2) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U46_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = x_2 + 2
POL( zeros ) = 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) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U105_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( U13_1(x_1) ) = 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) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2x_1 + x_3 + 2
POL( U132_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3
POL( U133_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 + x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_3 + x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = 0
POL( U33_1(x_1) ) = 2
POL( U96_1(x_1) ) = 0
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(339)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U41(X1, X2, X3)) -> MARK(X1)
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U42(X1, X2, X3)) -> MARK(X1)
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> MARK(X1)
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U44(X1, X2, X3)) -> MARK(X1)
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U45(X1, X2)) -> MARK(X1)
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(340) 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)) -> 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)
   ACTIVE(isNatIList(V)) -> MARK(U31(isNatIListKind(V), V))
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)) = x_1
   POL(U32(x_1, x_2)) = 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:

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


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

(341)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   MARK(U31(X1, X2)) -> ACTIVE(U31(mark(X1), X2))
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U31(X1, X2)) -> MARK(X1)
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(342) 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))
   MARK(U31(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( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U114_2(x_1, x_2) ) = 1
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( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = 2x_1
POL( U11_2(x_1, x_2) ) = 1
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = 2x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = 0
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = x_1 + 1
POL( U62_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatIListKind_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U31_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = x_1 + x_2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2x_2 + 2
POL( U106_1(x_1) ) = max{0, -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_1 + 2
POL( U122_1(x_1) ) = max{0, x_1 - 1}
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_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 + x_2 + x_4 - 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) ) = x_4 + 2
POL( take_2(x_1, x_2) ) = 0
POL( U33_1(x_1) ) = 2x_1 + 1
POL( U96_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(343)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(344) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U31(tt, V)) -> MARK(U32(isNatIListKind(V), V))
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( MARK_1(x_1) ) = max{0, -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) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 2
POL( U81_1(x_1) ) = 2x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( mark_1(x_1) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_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( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = 0
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( zeros ) = 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + 2x_3 + 2
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U106_1(x_1) ) = 2x_1 + 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) ) = 2x_2 + 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) ) = x_3 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2
POL( U135_4(x_1, ..., x_4) ) = 2x_1 + x_3 + 2x_4 + 1
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = 2x_1 + 1
POL( U33_1(x_1) ) = max{0, -2}
POL( U96_1(x_1) ) = max{0, -2}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(345)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> ACTIVE(U32(mark(X1), X2))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U32(X1, X2)) -> MARK(X1)
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(346) 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))
   MARK(U32(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( MARK_1(x_1) ) = max{0, x_1 - 1}
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) ) = 0
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1 + 2
POL( mark_1(x_1) ) = x_1
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_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( U52_1(x_1) ) = x_1 + 1
POL( U62_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatIListKind_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = x_3 + 2
POL( U103_3(x_1, ..., x_3) ) = x_2 + x_3
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 0
POL( U106_1(x_1) ) = 2
POL( s_1(x_1) ) = 2x_1 + 1
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, x_1 - 2}
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = x_2 + 2
POL( U132_4(x_1, ..., x_4) ) = x_1 + x_2 + 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + x_3 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_3 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U31_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U33_1(x_1) ) = x_1 + 1
POL( U96_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(347)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U32(tt, V)) -> MARK(U33(isNatList(V)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(348) 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)))
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( MARK_1(x_1) ) = max{0, -2}
POL( U114_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( U51_2(x_1, x_2) ) = 0
POL( U61_2(x_1, x_2) ) = 2
POL( U81_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 0
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 0
POL( U45_2(x_1, x_2) ) = 2
POL( U95_2(x_1, x_2) ) = 0
POL( length_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( zeros ) = 0
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) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + x_3
POL( U105_2(x_1, x_2) ) = x_1 + x_2 + 1
POL( U106_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U122_1(x_1) ) = 1
POL( U131_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = x_1 + x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2x_3 + 2
POL( take_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U31_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U32_2(x_1, x_2) ) = 2x_1 + x_2
POL( U33_1(x_1) ) = 2
POL( U96_1(x_1) ) = 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(349)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U33(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(350) 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) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = x_1 + 1
POL( U22_2(x_1, x_2) ) = 2x_1 + 1
POL( U41_3(x_1, ..., x_3) ) = 1
POL( U42_3(x_1, ..., x_3) ) = 1
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U11_2(x_1, x_2) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 0
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U52_1(x_1) ) = x_1 + 1
POL( U62_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatIListKind_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = 1
POL( U95_2(x_1, x_2) ) = 2x_1 + 1
POL( length_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2}
POL( U105_2(x_1, x_2) ) = x_2 + 2
POL( U106_1(x_1) ) = max{0, 2x_1 - 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) ) = 1
POL( U131_4(x_1, ..., x_4) ) = x_2 + x_3 + 2x_4 + 1
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = x_1 + 2
POL( U135_4(x_1, ..., x_4) ) = x_2 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, x_4 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U31_2(x_1, x_2) ) = x_2 + 2
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U33_1(x_1) ) = x_1 + 2
POL( U96_1(x_1) ) = x_1 + 1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(351)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U91(X1, X2, X3)) -> MARK(X1)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U92(X1, X2, X3)) -> MARK(X1)
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U93(X1, X2, X3)) -> MARK(X1)
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> MARK(X1)
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U95(X1, X2)) -> MARK(X1)
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(352) 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:

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


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

(353)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U21(X1, X2)) -> MARK(X1)
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U22(X1, X2)) -> MARK(X1)
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), V1))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(354) 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)
   ACTIVE(isNat(length(V1))) -> MARK(U11(isNatIListKind(V1), 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)) = 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)) = 1 + x_1
   POL(U22(x_1, x_2)) = 1 + 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)) = 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)) = 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)) = 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:

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


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

(355)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   MARK(U11(X1, X2)) -> ACTIVE(U11(mark(X1), X2))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(356) 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, -2}
POL( MARK_1(x_1) ) = max{0, x_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( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = x_1 + 1
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 1
POL( U92_3(x_1, ..., x_3) ) = x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 1
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U12_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( U52_1(x_1) ) = x_1 + 1
POL( U62_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = 1
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U11_2(x_1, x_2) ) = x_2 + 2
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = 1
POL( length_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( tt ) = 1
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_3 - 1}
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 1}
POL( U105_2(x_1, x_2) ) = x_1 + 2
POL( U106_1(x_1) ) = 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) ) = max{0, -2}
POL( U131_4(x_1, ..., x_4) ) = x_4 + 2
POL( U132_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U96_1(x_1) ) = 2x_1 + 1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U71(active(X)) -> U71(X)
   U71(mark(X)) -> U71(X)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(357)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), V1))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(358) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(U11(tt, V1)) -> MARK(U12(isNatIListKind(V1), 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, 2x_1 - 2}
POL( MARK_1(x_1) ) = 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) ) = 1
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 1
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 0
POL( U81_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( mark_1(x_1) ) = 2
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U12_2(x_1, x_2) ) = 2x_1 + x_2
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 1
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = 0
POL( length_1(x_1) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( zeros ) = 2
POL( 0 ) = 0
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 + 2x_2 - 2}
POL( U103_3(x_1, ..., x_3) ) = x_2 + 2x_3 + 1
POL( U104_3(x_1, ..., x_3) ) = x_1 + 2x_3
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U106_1(x_1) ) = 2
POL( U11_2(x_1, x_2) ) = x_1 + 2x_2 + 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, 2x_2 - 2}
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 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_3 + 2x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_4 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 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) ) = 0
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(359)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U61(X1, X2)) -> MARK(X1)
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(360) 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)))
   MARK(U61(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)) = 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)) = 1 + 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:

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


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

(361)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(362) 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))
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( MARK_1(x_1) ) = 1
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( U51_2(x_1, x_2) ) = 2
POL( U61_2(x_1, x_2) ) = 0
POL( U81_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( mark_1(x_1) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U52_1(x_1) ) = 0
POL( U71_1(x_1) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U23_1(x_1) ) = 0
POL( U45_2(x_1, x_2) ) = 2
POL( U95_2(x_1, x_2) ) = 2
POL( isNat_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( length_1(x_1) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( cons_2(x_1, x_2) ) = 2
POL( zeros ) = 0
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_2 + 2x_3 - 2}
POL( U103_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U105_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U106_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, -2}
POL( U13_1(x_1) ) = 1
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + x_4 + 1
POL( U133_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_3 + x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = max{0, x_1 - 1}
POL( take_2(x_1, x_2) ) = 0
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( U62_1(x_1) ) = 2
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(363)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U62(X)) -> MARK(X)
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(364) 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) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_1 - 1}
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) ) = 1
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 1
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U61_2(x_1, x_2) ) = max{0, -2}
POL( U81_1(x_1) ) = 2x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1
POL( mark_1(x_1) ) = x_1 + 2
POL( U112_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( U43_3(x_1, ..., x_3) ) = 0
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U52_1(x_1) ) = x_1 + 1
POL( U71_1(x_1) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatIListKind_1(x_1) ) = 0
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U23_1(x_1) ) = x_1 + 1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( length_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = 1
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 0
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) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( U105_2(x_1, x_2) ) = 2
POL( U106_1(x_1) ) = max{0, -2}
POL( U11_2(x_1, x_2) ) = x_2 + 2
POL( U12_2(x_1, x_2) ) = 2
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2x_2 + 2
POL( U122_1(x_1) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 + x_4 - 1}
POL( U132_4(x_1, ..., x_4) ) = 2x_1 + x_3 + x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 + x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 + 2x_4 - 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 - 2}
POL( take_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = x_1 + 2
POL( U96_1(x_1) ) = x_1 + 1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(365)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(take(V1, V2))) -> MARK(U61(isNatKind(V1), V2))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(366) 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_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)) = 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)) = 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:

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


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

(367)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   ACTIVE(isNatKind(length(V1))) -> MARK(U71(isNatIListKind(V1)))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(368) 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:

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


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

(369)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U71(X)) -> MARK(X)
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(370) 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( MARK_1(x_1) ) = max{0, x_1 - 1}
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) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = x_1
POL( U112_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( U52_1(x_1) ) = 2x_1 + 1
POL( U93_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U23_1(x_1) ) = x_1 + 1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( length_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = 0
POL( cons_2(x_1, x_2) ) = x_2 + 2
POL( zeros ) = 0
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U103_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = 2
POL( U106_1(x_1) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 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) ) = max{0, x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2
POL( U132_4(x_1, ..., x_4) ) = 2x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = x_1 + 2x_3 + 2
POL( U136_4(x_1, ..., x_4) ) = 2x_4 + 2
POL( take_2(x_1, x_2) ) = max{0, x_2 - 2}
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( U61_2(x_1, x_2) ) = 2
POL( U62_1(x_1) ) = 2
POL( U96_1(x_1) ) = 2x_1 + 1
POL( U71_1(x_1) ) = x_1 + 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(371)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U51(X1, X2)) -> MARK(X1)
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(372) 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:

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


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

(373)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   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))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(V2)))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(374) 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(U51(tt, V2)) -> MARK(U52(isNatIListKind(V2)))
   ACTIVE(U95(tt, V2)) -> MARK(U96(isNatList(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)) = x_3
   POL(U44(x_1, x_2, x_3)) = x_3
   POL(U45(x_1, x_2)) = 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)) = x_1
   POL(isNatIListKind(x_1)) = x_1
   POL(isNatKind(x_1)) = 0
   POL(isNatList(x_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:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(375)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   MARK(U42(X1, X2, X3)) -> ACTIVE(U42(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2))
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   MARK(U95(X1, X2)) -> ACTIVE(U95(mark(X1), X2))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(376) 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(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) ) = max{0, 2x_1 - 1}
POL( MARK_1(x_1) ) = 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( U41_3(x_1, ..., x_3) ) = 1
POL( U42_3(x_1, ..., x_3) ) = 0
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U81_1(x_1) ) = 0
POL( U91_3(x_1, ..., x_3) ) = 1
POL( U92_3(x_1, ..., x_3) ) = 1
POL( isNatKind_1(x_1) ) = 1
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U44_3(x_1, ..., x_3) ) = 1
POL( U93_3(x_1, ..., x_3) ) = 1
POL( U94_3(x_1, ..., x_3) ) = 1
POL( isNatIListKind_1(x_1) ) = 1
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U23_1(x_1) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 1
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 1
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 1
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( length_1(x_1) ) = 1
POL( U111_3(x_1, ..., x_3) ) = 1
POL( U43_3(x_1, ..., x_3) ) = 1
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( zeros ) = 0
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) ) = 2x_2 + 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
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 - 2}
POL( U106_1(x_1) ) = max{0, x_1 - 2}
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U12_2(x_1, x_2) ) = 2
POL( U13_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatList_1(x_1) ) = 1
POL( U121_2(x_1, x_2) ) = 2x_1 + 2
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = x_1 + 2
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + 2x_3 + 2x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 + x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = x_1 + 2
POL( U136_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( take_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U31_2(x_1, x_2) ) = max{0, -2}
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = max{0, -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, 2x_1 - 2}
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(377)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   MARK(U52(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   MARK(U96(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(378) 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)
   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) ) = 2
POL( MARK_1(x_1) ) = x_1 + 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( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U81_1(x_1) ) = x_1
POL( U91_3(x_1, ..., x_3) ) = 0
POL( U92_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1
POL( mark_1(x_1) ) = x_1
POL( U112_3(x_1, ..., x_3) ) = 0
POL( U44_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) ) = 2x_1
POL( isNatIListKind_1(x_1) ) = 0
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U23_1(x_1) ) = x_1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = 2x_1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1
POL( length_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = 2x_1
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = 2
POL( zeros ) = 0
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 - 2}
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U106_1(x_1) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U13_1(x_1) ) = 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, -2}
POL( U132_4(x_1, ..., x_4) ) = x_3 + 2x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = x_3 + 2x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = 0
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = max{0, x_1 - 2}
POL( U52_1(x_1) ) = x_1 + 1
POL( U61_2(x_1, x_2) ) = max{0, 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) ) = x_1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   isNatList(active(X)) -> isNatList(X)
   isNatList(mark(X)) -> isNatList(X)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(379)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   MARK(isNatList(X)) -> ACTIVE(isNatList(X))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(380) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(isNatList(X)) -> ACTIVE(isNatList(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( MARK_1(x_1) ) = max{0, x_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) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 0
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U81_1(x_1) ) = x_1 + 1
POL( U91_3(x_1, ..., x_3) ) = 1
POL( U92_3(x_1, ..., x_3) ) = 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = x_1 + 2
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_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( isNatIListKind_1(x_1) ) = 0
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U23_1(x_1) ) = x_1 + 1
POL( U45_2(x_1, x_2) ) = 1
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1 + 1
POL( length_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = 0
POL( U43_3(x_1, ..., x_3) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U103_3(x_1, ..., x_3) ) = x_1 + 2
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( U105_2(x_1, x_2) ) = 2
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( U11_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U12_2(x_1, x_2) ) = 2x_2 + 2
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U122_1(x_1) ) = 2x_1 + 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 + x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = x_3 + 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = 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_2 - 2}
POL( U33_1(x_1) ) = 2x_1 + 2
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U61_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( U62_1(x_1) ) = 0
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 2x_1 + 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(381)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(isNatKind(V1), V1, V2))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(382) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(isNatList(cons(V1, V2))) -> MARK(U91(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( MARK_1(x_1) ) = max{0, -2}
POL( U114_2(x_1, x_2) ) = 2
POL( U21_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 1}
POL( U51_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U81_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U112_3(x_1, ..., x_3) ) = 0
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( U45_2(x_1, x_2) ) = 0
POL( U95_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( length_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 2
POL( cons_2(x_1, x_2) ) = 1
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( tt ) = 1
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + 2x_3 + 1
POL( U105_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U106_1(x_1) ) = max{0, x_1 - 2}
POL( U11_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( U12_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 2x_1 + 1
POL( U121_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U122_1(x_1) ) = x_1 + 1
POL( U131_4(x_1, ..., x_4) ) = max{0, -2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_3 + 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_1 + x_2 + 2x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U31_2(x_1, x_2) ) = x_2 + 2
POL( U32_2(x_1, x_2) ) = x_1 + 2
POL( U33_1(x_1) ) = 2x_1 + 2
POL( U52_1(x_1) ) = 2
POL( U61_2(x_1, x_2) ) = 2x_2 + 2
POL( U62_1(x_1) ) = 1
POL( U96_1(x_1) ) = 1
POL( U71_1(x_1) ) = 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(383)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U41(tt, V1, V2)) -> MARK(U42(isNatKind(V1), V1, V2))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U91(tt, V1, V2)) -> MARK(U92(isNatKind(V1), V1, V2))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(384) 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))
   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( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U21_2(x_1, x_2) ) = x_2 + 2
POL( U22_2(x_1, x_2) ) = x_2 + 2
POL( U41_3(x_1, ..., x_3) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2x_1 + 2
POL( U81_1(x_1) ) = 2x_1 + 1
POL( U92_3(x_1, ..., x_3) ) = 2x_2 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1 + 2
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U112_3(x_1, ..., x_3) ) = 0
POL( U44_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U93_3(x_1, ..., x_3) ) = 2x_2 + 1
POL( U94_3(x_1, ..., x_3) ) = x_2 + 1
POL( isNatIListKind_1(x_1) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U23_1(x_1) ) = x_1 + 1
POL( U45_2(x_1, x_2) ) = x_2 + 2
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = x_1 + 1
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = x_1 + 1
POL( s_1(x_1) ) = x_1 + 1
POL( length_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = 1
POL( U43_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U91_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( cons_2(x_1, x_2) ) = x_2 + 1
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U106_1(x_1) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U13_1(x_1) ) = x_1
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) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = x_3 + 2x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = x_3 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 - 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) ) = max{0, 2x_1 - 2}
POL( U52_1(x_1) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = 1
POL( U62_1(x_1) ) = max{0, -2}
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(385)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   MARK(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   MARK(U91(X1, X2, X3)) -> ACTIVE(U91(mark(X1), X2, X3))
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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(U41(X1, X2, X3)) -> ACTIVE(U41(mark(X1), X2, X3))
   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, x_1 - 1}
POL( MARK_1(x_1) ) = 1
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) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U81_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, 2x_1 - 1}
POL( mark_1(x_1) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U23_1(x_1) ) = max{0, -2}
POL( U45_2(x_1, x_2) ) = 2
POL( U95_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( isNat_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U43_3(x_1, ..., x_3) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, -2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( zeros ) = 2
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = x_1 + 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = x_2 + x_3
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
POL( U105_2(x_1, x_2) ) = 2x_1
POL( U106_1(x_1) ) = max{0, -2}
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 - 1}
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) ) = 1
POL( U131_4(x_1, ..., x_4) ) = 2x_2 + 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = x_2 + x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_3 - 2}
POL( take_2(x_1, x_2) ) = 1
POL( U31_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U52_1(x_1) ) = 2x_1 + 2
POL( U61_2(x_1, x_2) ) = max{0, x_1 - 1}
POL( U62_1(x_1) ) = 0
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = max{0, -2}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   isNatIListKind(active(X)) -> isNatIListKind(X)
   isNatIListKind(mark(X)) -> isNatIListKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)
   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)


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

(387)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   MARK(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   MARK(U92(X1, X2, X3)) -> ACTIVE(U92(mark(X1), X2, X3))
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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(U43(X1, X2, X3)) -> ACTIVE(U43(mark(X1), X2, X3))
   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) ) = 0
POL( MARK_1(x_1) ) = max{0, x_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) ) = 0
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U81_1(x_1) ) = x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = 2x_1
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_3(x_1, ..., x_3) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = 1
POL( U94_3(x_1, ..., x_3) ) = 1
POL( isNatIListKind_1(x_1) ) = 0
POL( U113_3(x_1, ..., x_3) ) = 0
POL( U23_1(x_1) ) = x_1 + 1
POL( U45_2(x_1, x_2) ) = 0
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( length_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( U92_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
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) ) = 2x_2 + 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U106_1(x_1) ) = max{0, -2}
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U12_2(x_1, x_2) ) = x_2 + 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) ) = max{0, -2}
POL( U131_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_3 + x_4 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2x_3 + 2x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_2 + x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + x_4 - 2}
POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( U31_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U52_1(x_1) ) = max{0, x_1 - 2}
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) ) = max{0, x_3 - 2}
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = max{0, x_1 - 1}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   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)
   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)
   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)
   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)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(389)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   MARK(isNatIListKind(X)) -> ACTIVE(isNatIListKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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(isNatIListKind(X)) -> ACTIVE(isNatIListKind(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( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U114_2(x_1, x_2) ) = 2x_1 + 1
POL( U21_2(x_1, x_2) ) = 0
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U81_1(x_1) ) = x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = x_1
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U44_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) ) = 1
POL( isNatIListKind_1(x_1) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = 2x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1 + 1
POL( length_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = 2
POL( zeros ) = 0
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = x_3
POL( U103_3(x_1, ..., x_3) ) = x_1 + 2x_3 + 2
POL( U104_3(x_1, ..., x_3) ) = max{0, -2}
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U106_1(x_1) ) = 2
POL( U11_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( U12_2(x_1, x_2) ) = 2x_1 + x_2 + 2
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) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = x_1 + 2x_3 + 2
POL( U134_4(x_1, ..., x_4) ) = x_1 + x_2 + 2x_3 + 2x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( take_2(x_1, x_2) ) = max{0, 2x_2 - 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) ) = max{0, x_1 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2}
POL( U52_1(x_1) ) = 2x_1 + 2
POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( U96_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   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)
   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)
   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)
   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)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(391)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   ACTIVE(U43(tt, V1, V2)) -> MARK(U44(isNatIListKind(V2), V1, V2))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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(U43(tt, V1, V2)) -> MARK(U44(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( MARK_1(x_1) ) = 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) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U51_2(x_1, x_2) ) = max{0, -2}
POL( U81_1(x_1) ) = 0
POL( isNatKind_1(x_1) ) = 2
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = 2
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = 2
POL( length_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U12_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, -2}
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U52_1(x_1) ) = 0
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = max{0, -2}
POL( isNatIListKind_1(x_1) ) = 2
POL( U23_1(x_1) ) = 0
POL( U95_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = 2
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( zeros ) = 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, x_1 + x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U106_1(x_1) ) = 2
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U131_4(x_1, ..., x_4) ) = 2x_1 + 2x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 - 2}
POL( take_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U33_1(x_1) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 2x_1 + 2x_3 + 2
POL( U61_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U96_1(x_1) ) = x_1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   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)
   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)
   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)
   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)


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

(393)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(mark(X1), X2, X3))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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.

   MARK(U44(X1, X2, X3)) -> ACTIVE(U44(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( MARK_1(x_1) ) = max{0, x_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) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U51_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_1(x_1) ) = 2x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = x_1
POL( U112_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( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = x_1 + 1
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, -2}
POL( U11_2(x_1, x_2) ) = 2x_1 + 2
POL( U12_2(x_1, x_2) ) = x_1 + x_2 + 1
POL( U122_1(x_1) ) = 1
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_3 + x_4 - 2}
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U43_3(x_1, ..., x_3) ) = x_1 + 2
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2x_1 + 1
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2x_2 + 2
POL( U106_1(x_1) ) = max{0, -2}
POL( U13_1(x_1) ) = 2x_1 + 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U131_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + 2x_3 + x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = 2x_1 + 2x_3 + 2
POL( take_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( U61_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U96_1(x_1) ) = 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   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)
   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)
   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)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(395)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   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))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(V1), V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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.

   ACTIVE(U44(tt, V1, V2)) -> MARK(U45(isNat(V1), V2))
   ACTIVE(U92(tt, V1, V2)) -> MARK(U93(isNatIListKind(V2), V1, V2))
   ACTIVE(isNatIListKind(cons(V1, V2))) -> MARK(U51(isNatKind(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( MARK_1(x_1) ) = 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( U41_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U51_2(x_1, x_2) ) = x_1 + x_2
POL( U81_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = 1
POL( U112_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( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = max{0, -2}
POL( length_1(x_1) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2}
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2x_1 + 2
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = 2
POL( isNatIListKind_1(x_1) ) = 2x_1
POL( U23_1(x_1) ) = max{0, 2x_1 - 2}
POL( U95_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( cons_2(x_1, x_2) ) = 2x_2 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U105_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U106_1(x_1) ) = 2
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( U131_4(x_1, ..., x_4) ) = x_1 + x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = x_1 + 2
POL( U134_4(x_1, ..., x_4) ) = x_3 + x_4
POL( U135_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2}
POL( take_2(x_1, x_2) ) = x_2
POL( U33_1(x_1) ) = 2
POL( U42_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U61_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U91_3(x_1, ..., x_3) ) = x_1
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U96_1(x_1) ) = 1
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   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)
   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)
   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)


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

(397)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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(U93(X1, X2, X3)) -> ACTIVE(U93(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(ACTIVE(x_1)) = 0
   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)) = 0
   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
   POL(U45(x_1, x_2)) = 0
   POL(U46(x_1)) = x_1
   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)) = x_1
   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)) = 1
   POL(U94(x_1, x_2, x_3)) = 0
   POL(U95(x_1, x_2)) = 0
   POL(U96(x_1)) = 0
   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)) = 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:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   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)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(399)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U45(X1, X2)) -> ACTIVE(U45(mark(X1), X2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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.

   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( MARK_1(x_1) ) = max{0, x_1 - 1}
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) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U81_1(x_1) ) = x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = x_1
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = 1
POL( U45_2(x_1, x_2) ) = 2x_2 + 2
POL( length_1(x_1) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2
POL( U104_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2
POL( U11_2(x_1, x_2) ) = 2x_1 + 2
POL( U12_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = max{0, -2}
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_4 - 2}
POL( U31_2(x_1, x_2) ) = 2x_2 + 2
POL( U32_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = x_3 + 2
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = max{0, -2}
POL( U71_1(x_1) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = x_1 + 1
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = x_1 + 1
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 1
POL( cons_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( zeros ) = 2
POL( 0 ) = 2
POL( U101_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2
POL( tt ) = 1
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U106_1(x_1) ) = 2
POL( U13_1(x_1) ) = 2x_1 + 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = max{0, x_2 + x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2
POL( U134_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_3 + x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2x_3 + x_4 + 2
POL( take_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = 2
POL( U42_3(x_1, ..., x_3) ) = x_2 + 2
POL( U51_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U92_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U96_1(x_1) ) = 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   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)
   U46(active(X)) -> U46(X)
   U46(mark(X)) -> U46(X)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(401)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(V2)))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U93(tt, V1, V2)) -> MARK(U94(isNatIListKind(V2), V1, V2))
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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.

   ACTIVE(U45(tt, V2)) -> MARK(U46(isNatIList(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, x_1 - 2}
POL( MARK_1(x_1) ) = max{0, -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) ) = max{0, 2x_2 - 2}
POL( U81_1(x_1) ) = max{0, -2}
POL( isNatKind_1(x_1) ) = 2
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( mark_1(x_1) ) = max{0, 2x_1 - 2}
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U94_3(x_1, ..., x_3) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
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( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 + x_4 - 2}
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U32_2(x_1, x_2) ) = 2
POL( U43_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = max{0, x_1 - 1}
POL( U71_1(x_1) ) = 0
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + 1
POL( isNatIListKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = 2
POL( isNat_1(x_1) ) = 0
POL( U46_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2
POL( U105_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 1
POL( U121_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U131_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, x_2 + x_4 - 2}
POL( U134_4(x_1, ..., x_4) ) = 2x_3 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 - 2}
POL( U136_4(x_1, ..., x_4) ) = x_3 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( U45_2(x_1, x_2) ) = 2x_1 + 1
POL( U51_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U61_2(x_1, x_2) ) = max{0, 2x_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 + x_3 - 2}
POL( U96_1(x_1) ) = 2
POL( nil ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   isNatIList(active(X)) -> isNatIList(X)
   isNatIList(mark(X)) -> isNatIList(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   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)
   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)
   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)


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

(403)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   MARK(U46(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   MARK(U94(X1, X2, X3)) -> ACTIVE(U94(mark(X1), X2, X3))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(404) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(isNatIList(X)) -> ACTIVE(isNatIList(X))
   MARK(U46(X)) -> MARK(X)
   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) ) = max{0, -2}
POL( MARK_1(x_1) ) = max{0, x_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) ) = max{0, -2}
POL( U41_3(x_1, ..., x_3) ) = max{0, -2}
POL( U81_1(x_1) ) = 2x_1 + 1
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = 2x_1 + 2
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( U94_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( length_1(x_1) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = x_1 + x_2 + 2
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = 2
POL( U122_1(x_1) ) = max{0, x_1 - 1}
POL( U132_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2
POL( U31_2(x_1, x_2) ) = 2x_1 + 2
POL( U32_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U43_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U44_3(x_1, ..., x_3) ) = x_1 + 2x_3 + 2
POL( U52_1(x_1) ) = 0
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = 2
POL( U93_3(x_1, ..., x_3) ) = x_1 + 2
POL( isNatIListKind_1(x_1) ) = 2
POL( U23_1(x_1) ) = x_1 + 1
POL( U95_2(x_1, x_2) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( isNatIList_1(x_1) ) = 2
POL( s_1(x_1) ) = x_1 + 1
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = 2
POL( U105_2(x_1, x_2) ) = 2x_2 + 2
POL( U106_1(x_1) ) = 2
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2x_3 + x_4 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U135_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U136_4(x_1, ..., x_4) ) = x_3
POL( take_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U46_1(x_1) ) = x_1 + 2
POL( U51_2(x_1, x_2) ) = 2x_2 + 2
POL( U61_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U91_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
POL( U92_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( U96_1(x_1) ) = 2
POL( nil ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   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)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(405)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   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)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNatKind(V1), V1, V2))
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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.

   ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(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( MARK_1(x_1) ) = max{0, -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) ) = 2x_3 + 2
POL( U81_1(x_1) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = max{0, -2}
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 0
POL( U113_3(x_1, ..., x_3) ) = max{0, -2}
POL( length_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 + 2x_3 - 2}
POL( U11_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U12_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U122_1(x_1) ) = max{0, -2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_4 - 2}
POL( U31_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U32_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( U52_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = 0
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = 0
POL( U95_2(x_1, x_2) ) = 2
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = 2x_1 + 2
POL( zeros ) = 0
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) ) = 2
POL( U105_2(x_1, x_2) ) = 2
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2x_1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2x_2 + 2
POL( U131_4(x_1, ..., x_4) ) = x_1 + 2
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U134_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + x_3 + 2x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = x_2 + 2
POL( U136_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = 2x_2 + 2
POL( U33_1(x_1) ) = max{0, -2}
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U45_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U46_1(x_1) ) = max{0, x_1 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U61_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( U92_3(x_1, ..., x_3) ) = x_1 + 2
POL( U96_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   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)
   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)


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

(407)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   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)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(U94(tt, V1, V2)) -> MARK(U95(isNat(V1), V2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(408) 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, x_1 - 2}
POL( MARK_1(x_1) ) = 0
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( U81_1(x_1) ) = max{0, x_1 - 2}
POL( isNatKind_1(x_1) ) = 2
POL( active_1(x_1) ) = x_1 + 2
POL( mark_1(x_1) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 2
POL( U111_3(x_1, ..., x_3) ) = 2
POL( U113_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U104_3(x_1, ..., x_3) ) = 2
POL( U11_2(x_1, x_2) ) = 2
POL( U12_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U122_1(x_1) ) = max{0, -2}
POL( U132_4(x_1, ..., x_4) ) = x_2 + 2
POL( U31_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U32_2(x_1, x_2) ) = 1
POL( U43_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U52_1(x_1) ) = 0
POL( U62_1(x_1) ) = 0
POL( U71_1(x_1) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U94_3(x_1, ..., x_3) ) = 2x_1
POL( isNatIListKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2
POL( U95_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( isNat_1(x_1) ) = 2
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + 2x_3 - 2}
POL( tt ) = 2
POL( U102_3(x_1, ..., x_3) ) = max{0, x_2 - 1}
POL( U105_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 0
POL( U13_1(x_1) ) = 0
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 1
POL( U131_4(x_1, ..., x_4) ) = 2x_3 + 2
POL( U133_4(x_1, ..., x_4) ) = 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) ) = x_1 + x_3 + 2
POL( take_2(x_1, x_2) ) = 0
POL( U33_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U45_2(x_1, x_2) ) = 0
POL( U46_1(x_1) ) = 0
POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U61_2(x_1, x_2) ) = 2
POL( U91_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_3 - 2}
POL( U92_3(x_1, ..., x_3) ) = 2
POL( U96_1(x_1) ) = 0
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   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)
   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)


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

(409)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   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)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(isNatKind(s(V1))) -> MARK(U81(isNatKind(V1)))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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(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( MARK_1(x_1) ) = max{0, x_1 - 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( U81_1(x_1) ) = x_1 + 2
POL( isNatKind_1(x_1) ) = 2x_1 + 2
POL( active_1(x_1) ) = x_1
POL( mark_1(x_1) ) = 2x_1
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = 0
POL( U113_3(x_1, ..., x_3) ) = 2
POL( length_1(x_1) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = 2
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U11_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( U12_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U122_1(x_1) ) = max{0, -2}
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_2 - 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( U43_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3 + 2
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = max{0, x_1 + x_2 + x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 2
POL( U23_1(x_1) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 2
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( zeros ) = 1
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( tt ) = 1
POL( U102_3(x_1, ..., x_3) ) = x_1 + 2
POL( U105_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2x_1 + 2
POL( isNatList_1(x_1) ) = 2
POL( U121_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = max{0, x_2 + x_4 - 2}
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_3 - 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_2 + 2x_3 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = 2
POL( U41_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U45_2(x_1, x_2) ) = 2x_1 + 2
POL( U46_1(x_1) ) = 2
POL( U51_2(x_1, x_2) ) = 2x_1 + 2
POL( U61_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 1}
POL( U92_3(x_1, ..., x_3) ) = x_2 + 2
POL( U95_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U96_1(x_1) ) = max{0, 2x_1 - 1}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   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)
   U81(active(X)) -> U81(X)
   U81(mark(X)) -> U81(X)


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

(411)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(isNatKind(X)) -> ACTIVE(isNatKind(X))
   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   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)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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(isNatKind(X)) -> ACTIVE(isNatKind(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, 2x_1 - 1}
POL( MARK_1(x_1) ) = 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( isNatKind_1(x_1) ) = 0
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = 1
POL( U111_3(x_1, ..., x_3) ) = 1
POL( U113_3(x_1, ..., x_3) ) = 1
POL( length_1(x_1) ) = 1
POL( U103_3(x_1, ..., x_3) ) = x_2 + 2
POL( U104_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U122_1(x_1) ) = max{0, x_1 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 + x_4 - 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_1 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = x_1 + 2x_2 + 2x_3 + 2
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = max{0, -2}
POL( U93_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 1
POL( s_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
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 + x_3 - 1}
POL( U105_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(x_1) ) = 0
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2x_1 + 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_3 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 + x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( take_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_3 - 2}
POL( U42_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) ) = max{0, x_1 - 1}
POL( U61_2(x_1, x_2) ) = 2x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = 2
POL( U92_3(x_1, ..., x_3) ) = x_1 + x_2 + 2
POL( U95_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U96_1(x_1) ) = 2
POL( U81_1(x_1) ) = max{0, 2x_1 - 2}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNatKind(active(X)) -> isNatKind(X)
   isNatKind(mark(X)) -> isNatKind(X)
   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   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)
   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)


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

(413)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   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)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   MARK(U81(X)) -> MARK(X)
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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.

   MARK(U81(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, x_1 - 1}
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, -2}
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( active_1(x_1) ) = 2x_1 + 2
POL( mark_1(x_1) ) = 2x_1
POL( U113_3(x_1, ..., x_3) ) = 0
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( U103_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 + 2x_3 - 2}
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( U122_1(x_1) ) = max{0, -2}
POL( U132_4(x_1, ..., x_4) ) = 2
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = 2
POL( U43_3(x_1, ..., x_3) ) = x_1 + x_3 + 2
POL( U44_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3 + 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = 1
POL( U71_1(x_1) ) = 2
POL( U93_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = 2
POL( isNatIListKind_1(x_1) ) = 1
POL( U23_1(x_1) ) = x_1 + 1
POL( isNat_1(x_1) ) = 0
POL( U101_3(x_1, ..., x_3) ) = 2x_2 + x_3 + 2
POL( U102_3(x_1, ..., x_3) ) = x_3
POL( U134_4(x_1, ..., x_4) ) = x_1 + 2x_3 + x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = x_1 + x_2 + 2x_4 + 2
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 1}
POL( U51_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U61_2(x_1, x_2) ) = 2x_1 + 1
POL( U81_1(x_1) ) = x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U92_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1 + 1
POL( cons_2(x_1, x_2) ) = 2x_2 + 2
POL( zeros ) = 2
POL( 0 ) = 0
POL( tt ) = 0
POL( U105_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U106_1(x_1) ) = 0
POL( isNatIList_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) ) = max{0, 2x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = x_1 + x_2 + x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 + x_4 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = 1
POL( U45_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U46_1(x_1) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   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)


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

(415)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   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)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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.

   ACTIVE(U112(tt, L, N)) -> MARK(U113(isNat(N), L, N))
   ACTIVE(U21(tt, V1)) -> MARK(U22(isNatKind(V1), V1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(ACTIVE(x_1)) =  	[[0]] 	 +  	[[0, 1]] 	* 	x_1
>>>

   <<<
 POL(U111(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1 	 +  	[[0, 0], [1, 1]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(tt) =  	[[0], [1]]
>>>

   <<<
 POL(MARK(x_1)) =  	[[0]] 	 +  	[[0, 1]] 	* 	x_1
>>>

   <<<
 POL(U112(x_1, x_2, x_3)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 1]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(isNatIListKind(x_1)) =  	[[0], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(isNat(x_1)) =  	[[0], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(U113(x_1, x_2, x_3)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 1]] 	* 	x_2 	 +  	[[0, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U114(x_1, x_2)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(isNatKind(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[1], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(length(x_1)) =  	[[0], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U21(x_1, x_2)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(U22(x_1, x_2)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(U23(x_1)) =  	[[0], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(isNatList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(active(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(zeros) =  	[[0], [1]]
>>>

   <<<
 POL(0) =  	[[1], [1]]
>>>

   <<<
 POL(U101(x_1, x_2, x_3)) =  	[[0], [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], [0]] 	 +  	[[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], [0]] 	 +  	[[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)) =  	[[0], [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)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U106(x_1)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U11(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U12(x_1, x_2)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U13(x_1)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U121(x_1, x_2)) =  	[[1], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U122(x_1)) =  	[[1], [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 	 +  	[[1, 1], [0, 0]] 	* 	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 	 +  	[[0, 0], [1, 1]] 	* 	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], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U32(x_1, x_2)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(U33(x_1)) =  	[[0], [0]] 	 +  	[[0, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(U41(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [1, 1]] 	* 	x_3
>>>

   <<<
 POL(U42(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [1, 1]] 	* 	x_3
>>>

   <<<
 POL(U43(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 1], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U44(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 1], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U45(x_1, x_2)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U46(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(U52(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U62(x_1)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U91(x_1, x_2, x_3)) =  	[[0], [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)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U93(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U94(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U95(x_1, x_2)) =  	[[0], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U96(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U71(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U81(x_1)) =  	[[0], [1]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(nil) =  	[[1], [1]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


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

(417)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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(U112(X1, X2, X3)) -> ACTIVE(U112(mark(X1), X2, X3))
   MARK(U21(X1, X2)) -> ACTIVE(U21(mark(X1), X2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = 2
POL( U112_3(x_1, ..., x_3) ) = max{0, -2}
POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = 2
POL( active_1(x_1) ) = 2
POL( mark_1(x_1) ) = max{0, 2x_1 - 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) ) = 2
POL( length_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 - 1}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U11_2(x_1, x_2) ) = max{0, x_2 - 1}
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U122_1(x_1) ) = x_1 + 2
POL( U132_4(x_1, ..., x_4) ) = x_1 + 2
POL( U31_2(x_1, x_2) ) = 2x_1 + 2
POL( U32_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U43_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U52_1(x_1) ) = max{0, x_1 - 2}
POL( U62_1(x_1) ) = max{0, 2x_1 - 2}
POL( U71_1(x_1) ) = 2
POL( U93_3(x_1, ..., x_3) ) = 2x_1 + x_3 + 2
POL( U94_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( isNatIListKind_1(x_1) ) = 2x_1
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 2
POL( U101_3(x_1, ..., x_3) ) = x_1 + x_3 + 2
POL( U102_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U134_4(x_1, ..., x_4) ) = 2
POL( U136_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 + x_4 - 2}
POL( U41_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U42_3(x_1, ..., x_3) ) = 2
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U81_1(x_1) ) = 2
POL( U91_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 2
POL( U92_3(x_1, ..., x_3) ) = max{0, x_1 + 2x_2 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, x_1 - 2}
POL( cons_2(x_1, x_2) ) = 0
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 0
POL( U105_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U106_1(x_1) ) = max{0, -2}
POL( isNatIList_1(x_1) ) = 1
POL( U13_1(x_1) ) = 2
POL( isNatList_1(x_1) ) = 1
POL( U121_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U131_4(x_1, ..., x_4) ) = 2x_1 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 + 2x_4 - 2}
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
POL( take_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U33_1(x_1) ) = max{0, x_1 - 2}
POL( U45_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U46_1(x_1) ) = max{0, 2x_1 - 2}
POL( U95_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U96_1(x_1) ) = 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(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)
   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)
   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)


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

(419)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(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)) -> MARK(X)
   MARK(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   MARK(U23(X)) -> MARK(X)
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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.

   MARK(s(X)) -> MARK(X)
   MARK(U23(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U112_3(x_1, ..., x_3) ) = 0
POL( ACTIVE_1(x_1) ) = x_1
POL( U111_3(x_1, ..., x_3) ) = 0
POL( active_1(x_1) ) = 2x_1 + 2
POL( U113_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U114_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = x_2 + 2
POL( length_1(x_1) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = max{0, -2}
POL( mark_1(x_1) ) = x_1
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 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( U122_1(x_1) ) = 2
POL( U132_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + x_4 - 2}
POL( U31_2(x_1, x_2) ) = 2
POL( U32_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = x_2 + 2
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = max{0, 2x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = 2x_1 + 2
POL( isNat_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U101_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U102_3(x_1, ..., x_3) ) = x_2 + x_3 + 2
POL( U134_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2x_3 + 2
POL( U136_4(x_1, ..., x_4) ) = 2
POL( U41_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U81_1(x_1) ) = 2x_1 + 2
POL( U91_3(x_1, ..., x_3) ) = x_3 + 2
POL( U92_3(x_1, ..., x_3) ) = 2
POL( isNatKind_1(x_1) ) = 0
POL( s_1(x_1) ) = 2x_1 + 2
POL( cons_2(x_1, x_2) ) = x_1 + 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 0
POL( U105_2(x_1, x_2) ) = x_2 + 2
POL( U106_1(x_1) ) = max{0, 2x_1 - 2}
POL( isNatIList_1(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) ) = 2
POL( U131_4(x_1, ..., x_4) ) = 2
POL( U133_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_3 + x_4 + 2
POL( U135_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + x_3 + 2
POL( take_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, x_1 - 2}
POL( U45_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U46_1(x_1) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U96_1(x_1) ) = max{0, x_1 - 2}
POL( nil ) = 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   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)
   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)
   length(active(X)) -> length(X)
   length(mark(X)) -> length(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(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)
   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)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   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)
   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)


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

(421)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(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(length(X)) -> ACTIVE(length(mark(X)))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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(U113(X1, X2, X3)) -> ACTIVE(U113(mark(X1), X2, X3))
   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( MARK_1(x_1) ) = max{0, 2x_1 - 2}
POL( U112_3(x_1, ..., x_3) ) = 0
POL( ACTIVE_1(x_1) ) = 0
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( U113_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 + 2
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = x_2 + 1
POL( length_1(x_1) ) = 2
POL( U103_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( mark_1(x_1) ) = 2
POL( U104_3(x_1, ..., x_3) ) = max{0, x_2 - 1}
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 1}
POL( U12_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U132_4(x_1, ..., x_4) ) = x_2 + x_3 + 2
POL( U31_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U52_1(x_1) ) = 2
POL( U62_1(x_1) ) = 1
POL( U71_1(x_1) ) = 2
POL( U93_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = x_1 + 2
POL( isNatIListKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = 0
POL( isNat_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, -2}
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U102_3(x_1, ..., x_3) ) = max{0, x_3 - 2}
POL( U134_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_4 - 2}
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 - 2}
POL( U41_3(x_1, ..., x_3) ) = max{0, x_2 + x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 1
POL( U51_2(x_1, x_2) ) = x_2 + 2
POL( U61_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U81_1(x_1) ) = max{0, -2}
POL( U91_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 2
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( isNatKind_1(x_1) ) = 1
POL( s_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 0
POL( U105_2(x_1, x_2) ) = 2
POL( U106_1(x_1) ) = 2
POL( isNatIList_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, x_1 + x_2 - 2}
POL( U131_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 2x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = x_3 + 2
POL( U135_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2}
POL( take_2(x_1, x_2) ) = 2
POL( U33_1(x_1) ) = max{0, x_1 - 2}
POL( U45_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
POL( U46_1(x_1) ) = max{0, -2}
POL( U95_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U96_1(x_1) ) = max{0, -2}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


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

(423)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U113(tt, L, N)) -> MARK(U114(isNatKind(N), L))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

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

(424) 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( MARK_1(x_1) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( ACTIVE_1(x_1) ) = 2x_1
POL( U111_3(x_1, ..., x_3) ) = max{0, -2}
POL( active_1(x_1) ) = max{0, x_1 - 2}
POL( U114_2(x_1, x_2) ) = max{0, -2}
POL( U22_2(x_1, x_2) ) = max{0, -2}
POL( U103_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( mark_1(x_1) ) = max{0, -2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( U11_2(x_1, x_2) ) = x_2 + 1
POL( U12_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U122_1(x_1) ) = 2
POL( U132_4(x_1, ..., x_4) ) = 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( U43_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( U44_3(x_1, ..., x_3) ) = 2
POL( U52_1(x_1) ) = max{0, -2}
POL( U62_1(x_1) ) = 2
POL( U71_1(x_1) ) = 2x_1 + 2
POL( U93_3(x_1, ..., x_3) ) = 2x_2 + 2
POL( U94_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( isNatIListKind_1(x_1) ) = 2x_1
POL( U23_1(x_1) ) = max{0, x_1 - 2}
POL( isNat_1(x_1) ) = 0
POL( U21_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U101_3(x_1, ..., x_3) ) = x_1 + 2
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 - 2}
POL( U134_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 + 2
POL( U136_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2}
POL( U41_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U42_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2}
POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U61_2(x_1, x_2) ) = 2
POL( U81_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) ) = max{0, x_3 - 2}
POL( isNatKind_1(x_1) ) = 0
POL( s_1(x_1) ) = x_1 + 2
POL( length_1(x_1) ) = x_1 + 2
POL( cons_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( zeros ) = 0
POL( 0 ) = 0
POL( tt ) = 2
POL( U105_2(x_1, x_2) ) = x_1 + x_2 + 2
POL( U106_1(x_1) ) = max{0, x_1 - 1}
POL( isNatIList_1(x_1) ) = 0
POL( U113_3(x_1, ..., x_3) ) = max{0, x_1 - 1}
POL( U13_1(x_1) ) = max{0, -2}
POL( isNatList_1(x_1) ) = 0
POL( U121_2(x_1, x_2) ) = 2
POL( U131_4(x_1, ..., x_4) ) = x_3 + 2
POL( U133_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( U135_4(x_1, ..., x_4) ) = x_2 + 2
POL( take_2(x_1, x_2) ) = 2x_1 + 2
POL( U33_1(x_1) ) = 2
POL( U45_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U46_1(x_1) ) = 2
POL( U95_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U96_1(x_1) ) = max{0, -2}
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(X)
   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)


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

(425)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))

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)

Q is empty.
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.

   ACTIVE(length(cons(N, L))) -> MARK(U111(isNatList(L), L, N))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( MARK_1(x_1) ) = max{0, -2}
POL( U112_3(x_1, ..., x_3) ) = max{0, 2x_1 - 2}
POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2}
POL( U111_3(x_1, ..., x_3) ) = 2
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( U114_2(x_1, x_2) ) = 2
POL( U22_2(x_1, x_2) ) = 2
POL( U103_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( mark_1(x_1) ) = max{0, x_1 - 2}
POL( U104_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 2}
POL( U11_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( U12_2(x_1, x_2) ) = x_1 + x_2 + 1
POL( U122_1(x_1) ) = max{0, 2x_1 - 2}
POL( U132_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + 2x_3 - 2}
POL( U31_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U32_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2}
POL( U43_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + x_3 - 2}
POL( U44_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2}
POL( U52_1(x_1) ) = max{0, 2x_1 - 2}
POL( U62_1(x_1) ) = max{0, x_1 - 2}
POL( U71_1(x_1) ) = max{0, x_1 - 2}
POL( U93_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_2 + x_3 - 2}
POL( U94_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2}
POL( isNatIListKind_1(x_1) ) = 0
POL( U23_1(x_1) ) = max{0, -2}
POL( isNat_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, -2}
POL( length_1(x_1) ) = x_1 + 1
POL( cons_2(x_1, x_2) ) = 2
POL( zeros ) = 0
POL( 0 ) = 0
POL( U101_3(x_1, ..., x_3) ) = max{0, 2x_1 + 2x_3 - 2}
POL( tt ) = 0
POL( U102_3(x_1, ..., x_3) ) = max{0, 2x_1 + x_3 - 2}
POL( isNatKind_1(x_1) ) = 2x_1
POL( U105_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2}
POL( U106_1(x_1) ) = 2
POL( isNatIList_1(x_1) ) = 2x_1 + 2
POL( U113_3(x_1, ..., x_3) ) = x_3 + 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_1 + x_2 + 2
POL( U131_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 2x_4 + 2
POL( U133_4(x_1, ..., x_4) ) = 2x_1 + 2x_3 + 2
POL( U134_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_2 + 2x_3 + x_4 - 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) ) = x_1 + x_2 + x_3 + 2x_4 + 2
POL( take_2(x_1, x_2) ) = x_1 + x_2
POL( U21_2(x_1, x_2) ) = max{0, 2x_1 + 2x_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) ) = 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) ) = max{0, -2}
POL( U61_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
POL( U91_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( U92_3(x_1, ..., x_3) ) = max{0, 2x_2 + x_3 - 2}
POL( U95_2(x_1, x_2) ) = 2
POL( U96_1(x_1) ) = max{0, 2x_1 - 2}
POL( U81_1(x_1) ) = 2
POL( nil ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   isNat(active(X)) -> isNat(X)
   isNat(mark(X)) -> isNat(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)
   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)


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

(427)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(U111(tt, L, N)) -> MARK(U112(isNatIListKind(L), L, N))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), 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)

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

(428) 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))
   MARK(isNat(X)) -> ACTIVE(isNat(X))
   ACTIVE(U114(tt, L)) -> MARK(s(length(L)))
   MARK(U111(X1, X2, X3)) -> ACTIVE(U111(mark(X1), X2, X3))
   MARK(U114(X1, X2)) -> ACTIVE(U114(mark(X1), X2))
   ACTIVE(U22(tt, V1)) -> MARK(U23(isNat(V1)))
   MARK(U22(X1, X2)) -> ACTIVE(U22(mark(X1), X2))
   ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1), V1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE

MARK(x1)  =  x1

U112(x1, x2, x3)  =  U112

isNat(x1)  =  isNat

s(x1)  =  s

U111(x1, x2, x3)  =  U111

U114(x1, x2)  =  U114

U23(x1)  =  U23

U22(x1, x2)  =  U22

U21(x1, x2)  =  U21


Knuth-Bendix order [KBO] with precedence:trivial

and weight map:

   s=1
   isNat=10
   U114=10
   U22=10
   U112=5
   ACTIVE=9
   U21=5
   U23=5
   U111=10

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   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)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   U23(active(X)) -> U23(X)
   U23(mark(X)) -> U23(X)
   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)


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

(429)
Obligation:
Q DP problem:
P is empty.
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)

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

(430) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(431)
YES